/*
 * Java Genetic Algorithm Library (jenetics-7.1.0).
 * Copyright (c) 2007-2022 Franz Wilhelmstötter
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *
 * Author:
 *    Franz Wilhelmstötter (franz.wilhelmstoetter@gmail.com)
 */
package io.jenetics.ext.util;

import static java.lang.String.format;
import static java.util.Objects.requireNonNull;
import static java.util.Spliterators.spliteratorUnknownSize;
import static io.jenetics.internal.util.SerialIO.readIntArray;
import static io.jenetics.internal.util.SerialIO.writeIntArray;

import java.io.DataInput;
import java.io.DataOutput;
import java.io.IOException;
import java.io.InvalidObjectException;
import java.io.ObjectInputStream;
import java.io.Serial;
import java.io.Serializable;
import java.lang.reflect.Array;
import java.util.Arrays;
import java.util.Iterator;
import java.util.Objects;
import java.util.Optional;
import java.util.Spliterator;
import java.util.Spliterators;
import java.util.function.BiFunction;
import java.util.function.Function;
import java.util.stream.Stream;
import java.util.stream.StreamSupport;

import io.jenetics.util.ISeq;
import io.jenetics.util.Self;

/**
 * General purpose tree structure. The interface only contains tree read methods.
 * For a mutable tree implementation have a look at the {@link TreeNode} class.
 *
 * @see TreeNode
 *
 * @author <a href="mailto:franz.wilhelmstoetter@gmail.com">Franz Wilhelmstötter</a>
 * @version 7.0
 * @since 3.9
 */
public interface Tree<V, T extends Tree<V, T>> extends Self<T>, Iterable<T> {

        /* *************************************************************************
         * Basic (abstract) operations. All other tree operations can be derived
         * from these methods.
         **************************************************************************/

        /**
         * Return the value of the current {@code Tree} node. The value may be
         * {@code null}.
         *
         * @return the value of the current {@code Tree} node
         */
        V value();

        /**
         * Return the <em>parent</em> node of this tree node.
         *
         * @return the parent node, or {@code Optional.empty()} if this node is the
         *         root of the tree
         */
        Optional<T> parent();

        /**
         * Return the child node with the given index.
         *
         * @param index the child index
         * @return the child node with the given index
         * @throws IndexOutOfBoundsException  if the {@code index} is out of
         *         bounds ({@code [0, childCount())})
         */
        T childAt(final int index);

        /**
         * Return the number of children this tree node consists of.
         *
         * @return the number of children this tree node consists of
         */
        int childCount();


        /* *************************************************************************
         * Derived operations
         **************************************************************************/

        /**
         * Return an iterator of the children of this {@code Tree} node.
         *
         * @return an iterator of the children of this {@code Tree} node.
         */
        default Iterator<T> childIterator() {
                return new TreeChildIterator<V, T>(self());
        }

        /**
         * Return a forward-order stream of this node's children.
         *
         * @return a stream of children of {@code this} node
         */
        default Stream<T> childStream() {
                return StreamSupport.stream(
                        Spliterators.spliterator(
                                childIterator(),
                                childCount(),
                                Spliterator.SIZED | Spliterator.ORDERED
                        ),
                        false
                );
        }

        /**
         * Returns {@code true} if this node is the root of the tree.
         *
         * @return {@code true} if this node is the root of its tree, {@code false}
         *         otherwise
         */
        default boolean isRoot() {
                return parent().isEmpty();
        }

        /**
         * Returns the depth of the tree rooted at this node. The <i>depth</i> of a
         * tree is the longest distance from {@code this} node to a leaf. If
         * {@code this} node has no children, 0 is returned. This operation is much
         * more expensive than {@link #level()} because it must effectively traverse
         * the entire tree rooted at {@code this} node.
         *
         * @return the depth of the tree whose root is this node
         */
        default int depth() {
                final Iterator<T> it = breadthFirstIterator();

                T last = null;
                while (it.hasNext()) {
                        last = it.next();
                }

                assert last != null;
                return last.level() - level();
        }

        /**
         * Returns the number of levels above this node. The <i>level</i> of a tree
         * is the distance from the root to {@code this} node. If {@code this} node
         * is the root, returns 0.
         *
         * @return the number of levels above this node
         */
        default int level() {
                Optional<T> ancestor = Optional.of(self());
                int levels = 0;
                while ((ancestor = ancestor.flatMap(Tree::parent)).isPresent()) {
                        ++levels;
                }

                return levels;
        }

        /**
         * Returns the index of the specified child in this node's child array, or
         * {@code -1} if {@code this} node doesn't contain the given {@code child}.
         * This method performs a linear search and is O(n) where {@code n} is the
         * number of children.
         *
         * @param child  the TreeNode to search for among this node's children
         * @throws NullPointerException if the given {@code child} is {@code null}
         * @return the index of the node in this node's child array, or {@code -1}
         *         if the node could not be found
         */
        default int indexOf(final Tree<?, ?> child) {
                int index = -1;
                for (int i = 0, n = childCount(); i < n && index == -1; ++i) {
                        if (childAt(i).identical(child)) {
                                index = i;
                        }
                }

                return index;
        }

        /**
         * Return the number of nodes of {@code this} node (sub-tree).
         *
         * @return the number of nodes of {@code this} node (sub-tree)
         */
        default int size() {
                return Trees.countChildren(this) + 1;
        }

        /**
         * A tree is considered <em>empty</em> if it's {@link #value()} is
         * {@code null} and has no children and parent. A newly created tree node
         * with no value is <em>empty</em>.
         *
         * <pre>{@code
         * final Tree<String, ?> tree = TreeNode.of();
         * assert tree.isEmpty();
         * }</pre>
         *
         * @since 7.0
         *
         * @return {@code true} if {@code this} tree is empty, {@code false}
         *          otherwise
         */
        default boolean isEmpty() {
                return value() == null && childCount() == 0 && parent().isEmpty();
        }


        /* *************************************************************************
         * Query operations
         **************************************************************************/

        /**
         * Return the child node at the given {@code path}. A path consists of the
         * child index at a give level, starting with level 1. (The root note has
         * level zero.) {@code tree.childAtPath(Path.of(2))} will return the third
         * child node of {@code this} node, if it exists and
         * {@code tree.childAtPath(Path.of(2, 0))} will return the first child of
         * the third child of {@code this node}.
         *
         * @since 4.4
         *
         * @see #childAtPath(int...)
         *
         * @param path the child path
         * @return the child node at the given {@code path}
         * @throws NullPointerException if the given {@code path} array is
         *         {@code null}
         */
        default Optional<T> childAtPath(final Path path) {
                T node = self();
                for (int i = 0; i < path.length() && node != null; ++i) {
                        node = path.get(i) < node.childCount()
                                ? node.childAt(path.get(i))
                                : null;
                }

                return Optional.ofNullable(node);
        }

        /**
         * Return the child node at the given {@code path}. A path consists of the
         * child index at a give level, starting with level 1. (The root note has
         * level zero.) {@code tree.childAtPath(2)} will return the third child node
         * of {@code this} node, if it exists and {@code tree.childAtPath(2, 0)} will
         * return the first child of the third child of {@code this node}.
         *
         * @since 4.3
         *
         * @see #childAtPath(Path)
         *
         * @param path the child path
         * @return the child node at the given {@code path}
         * @throws NullPointerException if the given {@code path} array is
         *         {@code null}
         * @throws IllegalArgumentException if one of the path elements is smaller
         *         than zero
         */
        default Optional<T> childAtPath(final int... path) {
                return childAtPath(Path.of(path));
        }

        /**
         * Return {@code true} if the given {@code node} is an ancestor of
         * {@code this} node. This operation is at worst {@code O(h)} where {@code h}
         * is the distance from the root to {@code this} node.
         *
         * @param node the node to test
         * @return {@code true} if the given {@code node} is an ancestor of
         *         {@code this} node, {@code false} otherwise
         * @throws NullPointerException if the given {@code node} is {@code null}
         */
        default boolean isAncestor(final Tree<?, ?> node) {
                requireNonNull(node);

                Optional<T> ancestor = Optional.of(self());
                boolean result;
                do {
                        result = ancestor.filter(a -> a.identical(node)).isPresent();
                } while (!result &&
                                (ancestor = ancestor.flatMap(Tree::parent)).isPresent());

                return result;
        }

        /**
         * Return {@code true} if the given {@code node} is a descendant of
         * {@code this} node. If the given {@code node} is {@code null},
         * {@code false} is returned. This operation is at worst {@code O(h)} where
         * {@code h} is the distance from the root to {@code this} node.
         *
         * @param node the node to test as descendant of this node
         * @return {@code true} if this node is an ancestor of the given {@code node}
         * @throws NullPointerException if the given {@code node} is {@code null}
         */
        default boolean isDescendant(final Tree<?, ?> node) {
                return requireNonNull(node).isAncestor(this);
        }

        /**
         * Returns the nearest common ancestor to this node and the given {@code node}.
         * A node is considered an ancestor of itself.
         *
         * @param node {@code node} to find common ancestor with
         * @return nearest ancestor common to this node and the given {@code node},
         *         or {@link Optional#empty()} if no common ancestor exists.
         * @throws NullPointerException if the given {@code node} is {@code null}
         */
        default Optional<T> sharedAncestor(final T node) {
                requireNonNull(node);

                T ancestor = null;
                if (node.identical(this)) {
                        ancestor = self();
                } else {
                        final int level1 = level();
                        final int level2 = node.level();

                        T node1;
                        T node2;
                        int diff;
                        if (level2 > level1) {
                                diff = level2 - level1;
                                node1 = node;
                                node2 = self();
                        } else {
                                diff = level1 - level2;
                                node1 = self();
                                node2 = node;
                        }

                        while (diff > 0 && node1 != null) {
                                node1 = node1.parent().orElse(null);
                                --diff;
                        }

                        do {
                                if (node1 != null && node1.identical(node2)) {
                                        ancestor = node1;
                                }
                                node1 = node1 != null
                                        ? node1.parent().orElse(null)
                                        : null;
                                node2 = node2.parent().orElse(null);
                        } while (node1 != null && node2 != null && ancestor == null);
                }

                return Optional.ofNullable(ancestor);
        }

        /**
         * Returns true if and only if the given {@code node} is in the same tree as
         * {@code this} node.
         *
         * @param node the other node to check
         * @return true if the given {@code node} is in the same tree as {@code this}
         *         node, {@code false} otherwise.
         * @throws NullPointerException if the given {@code node} is {@code null}
         */
        default boolean isRelated(final Tree<?, ?> node) {
                requireNonNull(node);
                return node.root().identical(root());
        }

        /**
         * Returns the path from the root, to get to this node. The last element in
         * the path is this node.
         *
         * @since 5.1
         *
         * @return an array of TreeNode objects giving the path, where the
         *         first element in the path is the root and the last
         *         element is this node.
         */
        default ISeq<T> pathElements() {
                return Trees.pathElementsFromRoot(self(), 0).toISeq();
        }

        /**
         * Return the {@link Path} of {@code this} tree, such that
         * <pre>{@code
         * final Tree<Integer, ?> tree = ...;
         * final Tree.Path path = tree.path();
         * assert tree == tree.getRoot()
         *     .childAtPath(path)
         *     .orElse(null);
         * }</pre>
         *
         * @since 5.1
         *
         * @return the path from the root element to {@code this} node.
         */
        default Path path() {
                final int[] p = Trees.pathFromRoot(self(), 0);
                return Path.of(p);
        }

        /**
         * Returns the root of the tree that contains this node. The root is the
         * ancestor with no parent.
         *
         * @return the root of the tree that contains this node
         */
        default T root() {
                T anc = self();
                T prev;

                do {
                        prev = anc;
                        anc = anc.parent().orElse(null);
                } while (anc != null);

                return prev;
        }

        /* *************************************************************************
         * Child query operations
         **************************************************************************/

        /**
         * Return {@code true} if the given {@code node} is a child of {@code this}
         * node.
         *
         * @param node the other node to check
         * @return {@code true} if {@code node}is a child, {@code false} otherwise
         * @throws NullPointerException if the given {@code node} is {@code null}
         */
        default boolean isChild(final Tree<?, ?> node) {
                requireNonNull(node);
                return childCount() != 0 &&
                        node.parent().equals(Optional.of(self()));
        }

        /**
         * Return the first child of {@code this} node, or {@code Optional.empty()}
         * if {@code this} node has no children.
         *
         * @return the first child of this node
         */
        default Optional<T> firstChild() {
                return childCount() > 0
                        ? Optional.of(childAt(0))
                        : Optional.empty();
        }

        /**
         * Return the last child of {@code this} node, or {@code Optional.empty()}
         * if {@code this} node has no children.
         *
         * @return the last child of this node
         */
        default Optional<T> lastChild() {
                return childCount() > 0
                        ? Optional.of(childAt(childCount() - 1))
                        : Optional.empty();
        }

        /**
         * Return the child which comes immediately after {@code this} node. This
         * method performs a linear search of this node's children for {@code child}
         * and is {@code O(n)} where n is the number of children.
         *
         * @param child the child node
         * @return  the child of this node that immediately follows the {@code child},
         *          or {@code Optional.empty()} if the given {@code node} is the
         *          first node.
         * @throws NullPointerException if the given {@code child} is {@code null}
         */
        default Optional<T> childAfter(final Tree<?, ?> child) {
                requireNonNull(child);

                final int index = indexOf(child);
                if (index == -1) {
                        throw new IllegalArgumentException("The given node is not a child.");
                }

                return index < childCount() - 1
                        ? Optional.of(childAt(index + 1))
                        : Optional.empty();
        }

        /**
         * Return the child which comes immediately before {@code this} node. This
         * method performs a linear search of this node's children for {@code child}
         * and is {@code O(n)} where n is the number of children.
         *
         * @param child the child node
         * @return  the child of this node that immediately precedes the {@code child},
         *          or {@code null} if the given {@code node} is the first node.
         * @throws NullPointerException if the given {@code child} is {@code null}
         */
        default Optional<T> childBefore(final Tree<?, ?> child) {
                requireNonNull(child);

                final int index = indexOf(child);
                if (index == -1) {
                        throw new IllegalArgumentException("The given node is not a child.");
                }

                return index > 0
                        ? Optional.of(childAt(index - 1))
                        : Optional.empty();
        }

        /**
         * Return the node that follows {@code this} node in a pre-order traversal
         * of {@code this} tree node. Return {@code Optional.empty()} if this node
         * is the last node of the traversal. This is an inefficient way to traverse
         * the entire tree use an iterator instead.
         *
         * @see #preorderIterator
         * @return the node that follows this node in a pre-order traversal, or
         *        {@code Optional.empty()} if this node is last
         */
        default Optional<T> nextNode() {
                Optional<T> next = Optional.empty();

                if (childCount() == 0) {
                        T node = self();
                        while (node != null && (next = node.nextSibling()).isEmpty()) {
                                node = node.parent().orElse(null);
                        }
                } else {
                        next = Optional.of(childAt(0));
                }

                return next;
        }

        /**
         * Return the node that precedes this node in a pre-order traversal of
         * {@code this} tree node. Returns {@code Optional.empty()} if this node is
         * the first node of the traversal, the root of the tree. This is an
         * inefficient way to traverse the entire tree; use an iterator instead.
         *
         * @see #preorderIterator
         * @return the node that precedes this node in a pre-order traversal, or
         *         {@code Optional.empty()} if this node is the first
         */
        default Optional<T> previousNode() {
                Optional<T> node = Optional.empty();

                if (parent().isPresent()) {
                        final Optional<T> prev = previousSibling();
                        if (prev.isPresent()) {
                                node = prev.get().childCount() == 0
                                        ? prev
                                        : prev.map(Tree::lastLeaf);
                        } else {
                                node = parent();
                        }
                }

                return node;
        }

        /* *************************************************************************
         * Sibling query operations
         **************************************************************************/

        /**
         * Test if the given {@code node} is a sibling of {@code this} node.
         *
         * @param node node to test as sibling of this node
         * @return {@code true} if the {@code node} is a sibling of {@code this}
         *         node
         * @throws NullPointerException if the given {@code node} is {@code null}
         */
        default boolean isSibling(final Tree<?, ?> node) {
                return identical(requireNonNull(node)) ||
                        parent().equals(node.parent());
        }

        /**
         * Return the number of siblings of {@code this} node. A node is its own
         * sibling (if it has no parent or no siblings, this method returns
         * {@code 1}).
         *
         * @return the number of siblings of {@code this} node
         */
        default int siblingCount() {
                return parent().map(Tree::childCount).orElse(1);
        }

        /**
         * Return the next sibling of {@code this} node in the parent's children
         * array, or {@code null} if {@code this} node has no parent or it is the
         * last child of the paren. This method performs a linear search that is
         * {@code O(n)} where n is the number of children; to traverse the entire
         * array, use the iterator of the parent instead.
         *
         * @see #childStream()
         * @return the sibling of {@code this} node that immediately follows
         *         {@code this} node
         */
        default Optional<T> nextSibling() {
                return parent().flatMap(p -> p.childAfter(self()));
        }

        /**
         * Return the previous sibling of {@code this} node in the parent's children
         * list, or {@code Optional.empty()} if this node has no parent or is the
         * parent's first child. This method performs a linear search that is O(n)
         * where n is the number of children.
         *
         * @return the sibling of {@code this} node that immediately precedes this
         *         node
         */
        default Optional<T> previousSibling() {
                return parent().flatMap(p -> p.childBefore(self()));
        }


        /* *************************************************************************
         * Leaf query operations
         **************************************************************************/

        /**
         * Return {@code true} if {@code this} node has no children.
         *
         * @return {@code true} if {@code this} node has no children, {@code false}
         *         otherwise
         */
        default boolean isLeaf() {
                return childCount() == 0;
        }

        /**
         * Return the first leaf that is a descendant of {@code this} node; either
         * this node or its first child's first leaf. {@code this} node is returned
         * if it is a leaf.
         *
         * @see #isLeaf
         * @see  #isDescendant
         * @return the first leaf in the subtree rooted at this node
         */
        default T firstLeaf() {
                T leaf = self();
                while (!leaf.isLeaf()) {
                        leaf = leaf.firstChild().orElseThrow(AssertionError::new);
                }

                return leaf;
        }

        /**
         * Return the last leaf that is a descendant of this node; either
         * {@code this} node or its last child's last leaf. Returns {@code this}
         * node if it is a leaf.
         *
         * @see #isLeaf
         * @see #isDescendant
         * @return the last leaf in this subtree
         */
        default T lastLeaf() {
                T leaf = self();
                while (!leaf.isLeaf()) {
                        leaf = leaf.lastChild().orElseThrow(AssertionError::new);
                }

                return leaf;
        }

        /**
         * Returns the leaf after {@code this} node or {@code Optional.empty()} if
         * this node is the last leaf in the tree.
         * <p>
         * In order to determine the next node, this method first performs a linear
         * search in the parent's child-list in order to find the current node.
         * <p>
         * That implementation makes the operation suitable for short traversals
         * from a known position. But to traverse all the leaves in the tree, you
         * should use {@link #depthFirstIterator()} to iterator the nodes in the
         * tree and use {@link #isLeaf()} on each node to determine which are leaves.
         *
         * @see #depthFirstIterator
         * @see #isLeaf
         * @return return the next leaf past this node
         */
        default Optional<T> nextLeaf() {
                return nextSibling()
                        .map(Tree::firstLeaf)
                        .or(() -> parent().flatMap(Tree::nextLeaf));
        }

        /**
         * Return the leaf before {@code this} node or {@code null} if {@code this}
         * node is the first leaf in the tree.
         * <p>
         * In order to determine the previous node, this method first performs a
         * linear search in the parent's child-list in order to find the current
         * node.
         * <p>
         * That implementation makes the operation suitable for short traversals
         * from a known position. But to traverse all the leaves in the tree, you
         * should use {@link #depthFirstIterator()} to iterate the nodes in the tree
         * and use {@link #isLeaf()} on each node to determine which are leaves.
         *
         * @see #depthFirstIterator
         * @see #isLeaf
         * @return returns the leaf before {@code this} node
         */
        default Optional<T> previousLeaf() {
                return previousSibling()
                        .map(Tree::lastLeaf)
                        .or(() -> parent().flatMap(Tree::previousLeaf));
        }

        /**
         * Returns the total number of leaves that are descendants of this node.
         * If this node is a leaf, returns {@code 1}. This method is {@code O(n)},
         * where n is the number of descendants of {@code this} node.
         *
         * @see #isLeaf()
         * @return the number of leaves beneath this node
         */
        default int leafCount() {
                return (int)leaves().count();
        }

        /**
         * Return a stream of leaves that are descendants of this node.
         *
         * @since 7.0
         *
         * @return a stream of leaves that are descendants of this node
         */
        default Stream<T> leaves() {
                return breadthFirstStream().filter(Tree::isLeaf);
        }

        /* *************************************************************************
         * Tree traversing.
         **************************************************************************/

        /**
         * Return an iterator that traverses the subtree rooted at {@code this}
         * node in breadth-first order. The first node returned by the iterator is
         * {@code this} node.
         * <p>
         * Modifying the tree by inserting, removing, or moving a node invalidates
         * any iterator created before the modification.
         *
         * @see #depthFirstIterator
         * @return an iterator for traversing the tree in breadth-first order
         */
        default Iterator<T> breadthFirstIterator() {
                return new TreeNodeBreadthFirstIterator<>(self());
        }

        /**
         * Return an iterator that traverses the subtree rooted at {@code this}.
         * The first node returned by the iterator is {@code this} node.
         * <p>
         * Modifying the tree by inserting, removing, or moving a node invalidates
         * any iterator created before the modification.
         *
         * @see #breadthFirstIterator
         * @return an iterator for traversing the tree in breadth-first order
         */
        @Override
        default Iterator<T> iterator() {
                return breadthFirstIterator();
        }

        /**
         * Return a stream that traverses the subtree rooted at {@code this} node in
         * breadth-first order. The first node returned by the stream is
         * {@code this} node.
         *
         * @see #depthFirstIterator
         * @see #stream()
         * @return a stream for traversing the tree in breadth-first order
         */
        default Stream<T> breadthFirstStream() {
                return StreamSupport
                        .stream(spliteratorUnknownSize(breadthFirstIterator(), 0), false);
        }

        /**
         * Return a stream that traverses the subtree rooted at {@code this} node in
         * breadth-first order. The first node returned by the stream is
         * {@code this} node.
         *
         * @see #breadthFirstStream
         * @return a stream for traversing the tree in breadth-first order
         */
        default Stream<T> stream() {
                return breadthFirstStream();
        }

        /**
         * Return an iterator that traverses the subtree rooted at {@code this} node
         * in pre-order. The first node returned by the iterator is {@code this}
         * node.
         * <p>
         * Modifying the tree by inserting, removing, or moving a node invalidates
         * any iterator created before the modification.
         *
         * @see #postorderIterator
         * @return an iterator for traversing the tree in pre-order
         */
        default Iterator<T> preorderIterator() {
                return new TreeNodePreorderIterator<>(self());
        }

        /**
         * Return a stream that traverses the subtree rooted at {@code this} node
         * in pre-order. The first node returned by the stream is {@code this} node.
         * <p>
         * Modifying the tree by inserting, removing, or moving a node invalidates
         * any iterator created before the modification.
         *
         * @see #preorderIterator
         * @return a stream for traversing the tree in pre-order
         */
        default Stream<T> preorderStream() {
                return StreamSupport
                        .stream(spliteratorUnknownSize(preorderIterator(), 0), false);
        }

        /**
         * Return an iterator that traverses the subtree rooted at {@code this}
         * node in post-order. The first node returned by the iterator is the
         * leftmost leaf.  This is the same as a depth-first traversal.
         *
         * @see #depthFirstIterator
         * @see #preorderIterator
         * @return an iterator for traversing the tree in post-order
         */
        default Iterator<T> postorderIterator() {
                return new TreeNodePostorderIterator<>(self());
        }

        /**
         * Return a stream that traverses the subtree rooted at {@code this} node in
         * post-order. The first node returned by the iterator is the leftmost leaf.
         * This is the same as a depth-first traversal.
         *
         * @see #depthFirstIterator
         * @see #preorderIterator
         * @return a stream for traversing the tree in post-order
         */
        default Stream<T> postorderStream() {
                return StreamSupport
                        .stream(spliteratorUnknownSize(postorderIterator(), 0), false);
        }

        /**
         * Return an iterator that traverses the subtree rooted at {@code this} node
         * in depth-first order. The first node returned by the iterator is the
         * leftmost leaf. This is the same as a postorder traversal.
         * <p>
         * Modifying the tree by inserting, removing, or moving a node invalidates
         * any iterator created before the modification.
         *
         * @see #breadthFirstIterator
         * @see #postorderIterator
         * @return an iterator for traversing the tree in depth-first order
         */
        default Iterator<T> depthFirstIterator() {
                return postorderIterator();
        }

        /**
         * Return a stream that traverses the subtree rooted at {@code this} node in
         * depth-first. The first node returned by the iterator is the leftmost leaf.
         * This is the same as a post-order traversal.
         *
         * @see #depthFirstIterator
         * @see #preorderIterator
         * @return a stream for traversing the tree in post-order
         */
        default Stream<T> depthFirstStream() {
                return postorderStream();
        }

        /**
         * Return an iterator that follows the path from {@code ancestor} to
         * {@code this} node. The iterator return {@code ancestor} as first element,
         * The creation of the iterator is O(m), where m is the number of nodes
         * between {@code this} node and the {@code ancestor}, inclusive.
         * <p>
         * Modifying the tree by inserting, removing, or moving a node invalidates
         * any iterator created before the modification.
         *
         * @see #isAncestor
         * @see #isDescendant
         * @param ancestor the ancestor node
         * @return an iterator for following the path from an ancestor of {@code this}
         *         node to this one
         * @throws IllegalArgumentException if the {@code ancestor} is not an
         *         ancestor of this node
         * @throws NullPointerException if the given {@code ancestor} is {@code null}
         */
        default Iterator<T> pathFromAncestorIterator(final Tree<?, ?> ancestor) {
                return new TreeNodePathIterator<>(ancestor, self());
        }

        /**
         * Return the path of {@code this} child node from the root node. You will
         * get {@code this} node, if you call {@link #childAtPath(Path)} on the
         * root node of {@code this} node.
         * <pre>{@code
         * final Tree<?, ?> node = ...;
         * final Tree<?, ?> root = node.getRoot();
         * final int[] path = node.childPath();
         * assert node == root.childAtPath(path);
         * }</pre>
         *
         * @since 4.4
         *
         * @see #childAtPath(Path)
         *
         * @return the path of {@code this} child node from the root node.
         */
        default Path childPath() {
                final Iterator<T> it = pathFromAncestorIterator(root());
                final int[] path = new int[level()];

                T tree = null;
                int index = 0;
                while (it.hasNext()) {
                        final T child = it.next();
                        if (tree != null) {
                                path[index++] = tree.indexOf(child);
                        }

                        tree = child;
                }

                assert index == path.length;

                return new Path(path);
        }

        /**
         * Tests whether {@code this} node is the same as the {@code other} node.
         * The default implementation returns the object identity,
         * {@code this == other}, of the two objects, but other implementations may
         * use different criteria for checking the <i>identity</i>.
         *
         * @param other the {@code other} node
         * @return {@code true} if the {@code other} node is the same as {@code this}
         *         node.
         */
        default boolean identical(final Tree<?, ?> other) {
                return this == other;
        }

        /**
         * Performs a reduction on the elements of {@code this} tree, using an
         * associative reduction function. This can be used for evaluating a given
         * expression tree in pre-order.
         * <pre>{@code
         * final Tree<String, ?> formula = TreeNode.parse("add(sub(6,div(230,10)),mul(5,6))");
         * final double result = formula.reduce(new Double[0], (op, args) ->
         *     switch (op) {
         *         case "add" -> args[0] + args[1];
         *         case "sub" -> args[0] - args[1];
         *         case "mul" -> args[0] * args[1];
         *         case "div" -> args[0] / args[1];
         *         default -> Double.parseDouble(op);
         *     }
         * );
         * assert result == 13.0;
         * }</pre>
         *
         * @since 7.1
         *
         * @param neutral the neutral element of the reduction. In most cases this will
         *        be {@code new U[0]}.
         * @param reducer the reduce function
         * @param <U> the result type
         * @return the result of the reduction, or {@code null} if {@code this} tree
         *         is empty ({@code isEmpty() == true})
         */
        default <U> U reduce(
                final U[] neutral,
                final BiFunction<? super V, ? super U[], ? extends U> reducer
        ) {
                requireNonNull(neutral);
                requireNonNull(reducer);

                @SuppressWarnings("unchecked")
                final class Reducing {
                        private U reduce(final Tree<V, ?> node) {
                                return node.isLeaf()
                                        ? reducer.apply(node.value(), neutral)
                                        : reducer.apply(node.value(), children(node));
                        }
                        private U[] children(final Tree<V, ?> node) {
                                final U[] values = (U[])Array.newInstance(
                                        neutral.getClass().getComponentType(),
                                        node.childCount()
                                );
                                for (int i = 0; i < node.childCount(); ++i) {
                                        values[i] = reduce(node.childAt(i));
                                }
                                return values;
                        }
                }

                return isEmpty() ? null : new Reducing().reduce(this);
        }


        /* *************************************************************************
         * 'toString' methods
         **************************************************************************/

        /**
         * Return a compact string representation of the given tree. The tree
         * <pre>
         *  mul
         *  ├── div
         *  │   ├── cos
         *  │   │   └── 1.0
         *  │   └── cos
         *  │       └── π
         *  └── sin
         *      └── mul
         *          ├── 1.0
         *          └── z
         *  </pre>
         * is printed as
         * <pre>
         *  mul(div(cos(1.0),cos(π)),sin(mul(1.0,z)))
         * </pre>
         *
         * @since 4.3
         *
         * @see #toParenthesesString()
         * @see TreeFormatter#PARENTHESES
         *
         * @param mapper the {@code mapper} which converts the tree value to a string
         * @return the string representation of the given tree
         */
        default String toParenthesesString(final Function<? super V, String> mapper) {
                return TreeFormatter.PARENTHESES.format(this, mapper);
        }

        /**
         * Return a compact string representation of the given tree. The tree
         * <pre>
         *  mul
         *  ├── div
         *  │   ├── cos
         *  │   │   └── 1.0
         *  │   └── cos
         *  │       └── π
         *  └── sin
         *      └── mul
         *          ├── 1.0
         *          └── z
         *  </pre>
         * is printed as
         * <pre>
         *  mul(div(cos(1.0), cos(π)), sin(mul(1.0, z)))
         * </pre>
         *
         * @since 4.3
         *
         * @see #toParenthesesString(Function)
         * @see TreeFormatter#PARENTHESES
         *
         * @return the string representation of the given tree
         * @throws NullPointerException if the {@code mapper} is {@code null}
         */
        default String toParenthesesString() {
                return toParenthesesString(Objects::toString);
        }

        /* *************************************************************************
         * Static helper methods.
         **************************************************************************/

        /**
         * Calculates the hash code of the given tree.
         *
         * @param tree the tree where the hash is calculated from
         * @return the hash code of the tree
         * @throws NullPointerException if the given {@code tree} is {@code null}
         */
        static int hashCode(final Tree<?, ?> tree) {
                return tree != null
                        ? tree.breadthFirstStream()
                                .mapToInt(node -> 31*Objects.hashCode(node.value()) + 37)
                                .sum() + 17
                        : 0;
        }

        /**
         * Checks if the two given trees has the same structure with the same values.
         *
         * @param a the first tree
         * @param b the second tree
         * @return {@code true} if the two given trees are structurally equals,
         *         {@code false} otherwise
         */
        static boolean equals(final Tree<?, ?> a, final Tree<?, ?> b) {
                return Trees.equals(a, b);
        }

        /**
         * Return a string representation of the given tree, like the following
         * example.
         *
         * <pre>
         *  mul(div(cos(1.0), cos(π)), sin(mul(1.0, z)))
         * </pre>
         *
         * This method is intended to be used when override the
         * {@link Object#toString()} method.
         *
         * @param tree the input tree
         * @return the string representation of the given tree
         */
        static String toString(final Tree<?, ?> tree) {
                return tree.toParenthesesString();
        }


        /* *************************************************************************
         * Inner classes
         **************************************************************************/

        /**
         * This class represents the path to child within a given tree. It allows
         * pointing (and fetch) a tree child.
         *
         * @see Tree#childAtPath(Path)
         *
         * @author <a href="mailto:franz.wilhelmstoetter@gmail.com">Franz Wilhelmstötter</a>
         * @version 6.0
         * @since 4.4
         */
        final class Path implements Serializable {

                @Serial
                private static final long serialVersionUID = 1L;

                private final int[] _path;

                private Path(final int[] path) {
                        _path = requireNonNull(path);
                }

                /**
                 * Return the path length, which is the level of the child {@code this}
                 * path points to.
                 *
                 * @return the path length
                 */
                public int length() {
                        return _path.length;
                }

                /**
                 * Return the child index at the given index (child level).
                 *
                 * @param index the path index
                 * @return the child index at the given child level
                 * @throws IndexOutOfBoundsException if the index is not with the range
                 *         {@code [0, length())}
                 */
                public int get(final int index) {
                        return _path[index];
                }

                /**
                 * Return the path as {@code int[]} array.
                 *
                 * @return the path as {@code int[]} array
                 */
                public int[] toArray() {
                        return _path.clone();
                }

                /**
                 * Appends the given {@code path} to {@code this} one.
                 *
                 * @param path the path to append
                 * @return a new {@code Path} with the given {@code path} appended
                 * @throws NullPointerException if the given {@code path} is {@code null}
                 */
                public Path append(final Path path) {
                        final int[] p = new int[length() + path.length()];
                        System.arraycopy(_path, 0, p, 0, length());
                        System.arraycopy(path._path, 0, p, length(), path.length());
                        return new Path(p);
                }

                @Override
                public int hashCode() {
                        return Arrays.hashCode(_path);
                }

                @Override
                public boolean equals(final Object obj) {
                        return obj == this ||
                                obj instanceof Path other &&
                                Arrays.equals(_path, other._path);
                }

                @Override
                public String toString() {
                        return Arrays.toString(_path);
                }

                /**
                 * Create a new path object from the given child indexes.
                 *
                 * @param path the child indexes
                 * @return a new tree path
                 * @throws IllegalArgumentException if one of the path elements is
                 *         smaller than zero
                 */
                public static Path of(final int... path) {
                        for (int i = 0; i < path.length; ++i) {
                                if (path[i] < 0) {
                                        throw new IllegalArgumentException(format(
                                                "Path element at position %d is smaller than zero: %d",
                                                i, path[i]
                                        ));
                                }
                        }

                        return new Path(path.clone());
                }


                /* *********************************************************************
                 *  Java object serialization
                 * ********************************************************************/

                @Serial
                private Object writeReplace() {
                        return new SerialProxy(SerialProxy.TREE_PATH, this);
                }

                @Serial
                private void readObject(final ObjectInputStream stream)
                        throws InvalidObjectException
                {
                        throw new InvalidObjectException("Serialization proxy required.");
                }


                void write(final DataOutput out) throws IOException {
                        writeIntArray(_path, out);
                }

                static Object read(final DataInput in) throws IOException {
                        return Path.of(readIntArray(in));
                }

        }

}