Binary Tree Definitions

GT Ch 2.3.3–2.3.4
A binary tree is either empty or nonempty, but not both.
A nonempty binary tree B consists of three things:
1. a node r called root
2. a left binary tree L called left subtree
3. a right binary tree R called right subtree
We write (r, L, R) to denote the nonempty binary tree B with root r, left subtree L, and right subtree R.
We assume no node is the root of more than one binary tree, and no binary tree is the subtree of itself.

Representing Binary Trees

Each node in a binary tree B contains
- a key field,
- a left field pointing to the left subtree
- a right field pointing to the right subtree
This allows traversing down the tree.
If traversing up the tree is also desired, the node should have another field p pointing to the parent node.

A node x is a right child of node y if some binary tree B has y as its root, and the right subtree of B has x as its root.
We define left child, child, and parent in the obvious way.
A node with no children is a leaf; a nonleaf node is called an internal node.
Node x is an ancestor of node y if either x = y, or x is the parent of an ancestor of y.
Node x is a proper ancestor of node y if x is an ancestor of y and x ≠ y.
We define descendant, and proper descendant in the obvious way.
A path is a sequence of nodes, each node after the first being the parent or child of the previous node. A path is simple if it contains no repeated node.

An edge is a pair of nodes (x, y) such that x is the parent of y or vice versa.
The length of a path is the number of edges it contains.
Consider a nonempty binary tree B = (r, L, R). Denote the depth of a node x in B by depth(x, B). We have \[ \mbox{depth}(x, B) = \left\{ \begin{array}{ll} 0, & \mbox{if $x = r$} \\ 1 + \mbox{depth}(x, L), & \mbox{if $x$ is in $L$} \\ 1 + \mbox{depth}(x, R), & \mbox{if $x$ is in $R$} \end{array} \right. \]
The height of B equals the largest depth of a node. It also equals the length of a longest simple path from r to a leaf.
All the nodes of depth $\ell$ are collectively called level $\ell$ of B.
A binary tree is full if all of its internal nodes have two children.

Full Binary Tree Lemma: The number of internal nodes in any nonempty full binary tree is one less than the number of leaves in the same tree.
Proof: By induction or counting the same thing two ways.
A binary tree is complete if it is full and all its leaves have the same depth.
A complete binary tree of height h has $1+2+\cdots+2^i+\cdots+2^h=2^{h+1}-1$ nodes.
Binary Tree Height Lemma: Any binary tree of n nodes has height $\ge \log (n+1)-1$.
Proof Sketch: This follows from $n\le 2^{h+1}-1$.

A rooted tree T is either empty or consists of a node called the root of T, and zero or more rooted trees called the subtrees of the root.
An ordered tree is a rooted tree where the subtrees are presented in a specific order.
All notions on binary trees (e.g., child, ancestor, depth, edge) extend to rooted trees.
Every ordered tree can be represented as a binary tree by using the left-child right-sibling representation.
For any integer $k\ge3$, the $k$-ary trees generalize the binary trees. A ternary tree is 3-ary.