Tree Concepts

Internal

• Trees
• Data Structures
• Graphs

Overview

A tree is a particular case of a graph.

Free Tree

A free tree is a connected, acyclic, undirected graph.

For an undirected graph G = (V, E), the following statements are equivalent:

1. G is a free tree.
2. Any two vertices in G are connected by a unique simple path.
3. G is connected, but if any edge is removed from E, the resulting graph is disconnected.
4. G is connected and │E│ = │V│ - 1.
5. G is acyclic and │E│ = │V│ - 1.
6. G is acyclic, but if any edge is added to E, the resulting graph contains a cycle.

Forest

If the undirected graph is acyclic, but disconnected, it is a forest.

Rooted Tree

A rooted tree is a free tree in which one of the vertices is distinguished from the others.

Root

We call the distinguished vertex the root of the tree.

Node

We often refer to the vertex of a rooted tree as a node of the tree.

Ancestors and Descendants

For a node x in a rooted tree, we call any node y on the simple path from root to x an ancestor of x. If y is an ancestor of x, then x is a descendant of y. Every node is both an ancestor and descendant of itself. If y is an ancestor of x and x ≠ y, then we call y a proper ancestor of x, and x is a proper descendant of y.

Subtree

The subtree rooted at x is the tree induced by descendants of x, rooted at x.

Parents, Children and Siblings

If the last edge on the simple path from the root of the tree to a node x is (y, x), then y is the parent of x and x is the child of y. The root is the only node in the tree with no parent. If two nodes have the same parent, they are siblings.

Leaf

A node with no children is a leaf or an external node. A non-leaf node is an internal node.

Depth and Height

Depth

The length of the simple path from the root to a node x, measured in the number of edges, is called the depth of the node x in the tree. The depth is always relative to the root.

Level

A level of a tree consists of all nodes at the same depth.

Height

The height of a node in a tree is the number of edges on the longest simple downward path from the node in question to a leaf.

The height of the tree is the height of its root. The height of the tree is also equal to the largest depth of any node in the tree.

The height is always relative to the lowest leaf.

Degree

Note that the degree of a node depends on whether we consider the tree to be a rooted tree or a free tree.

In case of a free tree, the degree of a node is the degree of a vertex in an undirected graph, defined as the number of adjacent vertices.

For a rooted tree, the degree is the number of children. The parent of the node does not count toward its degree.

Ordered Tree

An ordered tree is a rooted tree in which the children of each node are ordered. If a node has k children, then there is a first child, a second child, and the k^th child. Also see Difference between Binary Trees and Ordered Trees.

Positional Tree

In a positional tree, the children of a node are labeled with distinct positive integers. The i^th child of a node is absent if no child is labeled with integer i.

K-ary Tree

A k-ary tree is a positional tree in which for every node, all children with labels greater than k are absent.

A complete k-ary tree is a k-ary tree in which all leaves have the same depth and all internal nodes have degree k.

K-ary Tree Arithmetic

A complete k-ary tree of height h has k⁰ + k¹⁺ + ... + k^h-1 = (k^h - 1)/(k - 1) nodes and k^h leaves.

The height of a complete k-ary tree with n leaves is log_kn.

Types of K-ary Trees

B and B+ Trees

B and B+ Trees

Search Tree

• Binary search tree
• B tree

Binary Tree

A binary tree is defined recursively as a structure defined on a finite set of nodes that either:

• contains no nodes
• is composed of three disjoint sets of nodes: a root node, a binary tree called its left subtree and a binary tree called its right subtree.

If the left subtree is not empty, its root is called the left child of the root of the entire tree. If the right subtree is not empty, its root is called the right child of the root of the entire tree.

A binary tree can also be defined as a k-ary tree with k=2.

Types of Binary Trees

Full Binary Tree

A full binary tree, sometimes referred to as a proper or plane binary tree, is a tree in which every node has either 0 or 2 children. No node can have just one child. A full binary tree can also be defined with the recursive definition: a full binary tree is either a single node or a tree whose node has two subtrees, both of which are fully binary trees.

Complete Binary Tree

A complete binary tree is a tree in which every level, except possibly the last, is completely filled and all nodes in the last level are as far left as possible. These tree are sometimes referred to as almost complete, nearly complete, or as complete as possible. A complete binary tree can be efficiently represented using an array.

Balanced Binary Tree

Complete Binary Tree

A full binary tree, or a complete binary tree, is a binary tree where each node is either a leaf, or has a degree of exactly 2 - there are no degree-1 nodes.

The size of a complete binary tree as a function of its depth d is 2^d-1 nodes.

As Complete as Possible

There is also the concept of a tree "as complete as possible", or "as balanced as possible", where the all the layers in the tree are complete, with exception of the highest layer, which may be incomplete. This property of binary trees is relevant when building heaps. Usually, "as balanced as possible" means balanced enough to ensure O(n log n) for INSERT and SEARCH.

Full Binary Tree

Perfect Binary Tree

Difference between Binary Trees and Ordered Trees

A binary tree isn't simply an ordered tree with a degree of at most 2. In an ordered tree, there is no distinguishing a sole child as being either left or right. For a binary tree, the left or right position matters, so from this perspective, the concept of binary tree is stronger than the one of ordered tree. Binary trees can be implemented using ordered trees, if we maintain the position information necessary to a binary tree by replacing each missing child in the binary tree with a node having no children.

Binary Search Tree

These include balanced binary search trees, red-black trees, AVL trees, splay trees:

Binary Search Trees

Radix Trees (Tries)

Tries

Heaps

Heaps

Tree Walking

Tree Walking