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Mathematical properties of Binary Trees

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• Binary Trees
  • We focus on Binary Trees because they are used very frequently

  • Understanding the properties of Binary Trees will be a basis (groundwork) for understanding other trees.

• Binary Trees: Property 1

  

  A binary tree with N internal nodes has exactly N+1 external (leaf) nodes Proof: by induction Base case: Clearly, a binary tree with 0 internal nodes has exactly 1 external (leaf) node (see figure above) Induction step: Let us assume that it is true that a binary tree with m < N internal nodes has exactly m+1 external (leaf) nodes We must prove that a binary tree with N internal nodes has exactly N+1 external (leaf) nodes A binary tree with N internal nodes is made up of: A root (internal node) A left binary trees with k internal nodes (we don't care what value k has, we can handle it) A right binary trees with (N - 1 - k) internal nodes According to the induction assumption: The left binary trees with k internal nodes has exactly k+1 external nodes The right binary trees with (N - 1 - k) internal nodes has exactly (N - k) external nodes So the total number of external nodes in a binary tree with N internal nodes is: k+1 + (N - k) = N + 1 external nodes... see, I told you we can handle any k+1 :-)

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• Binary Trees: Property 2

  

  A binary tree with N internal nodes (and by Property 1, N+1 external nodes) has exactly 2N edges: Exactly N-1 edges connect 2 internal nodes Exactly N+1 edges connect an internal node and an external (leaf) node Proof: Fact: in a rooted tree, every node - except the root - has exactly 1 parent node There are N+1 external nodes, so there are N+1 edges connecting an internal node and an external (leaf) node There are N-1 internal nodes minus the root node, so there are N-1 edges connecting two internal nodes

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• Definition: Height and Level

  

  • The level of a vertex is number of edges from the node to the root (the level of the root node is 0)

  • The height of a tree is maximum level in the tree

  • The path length of a tree is the sum of all lengths of all paths from each node to the root

  • The internal path length of a tree is the sum of all lengths of all paths from each internal node to the root

  • The external path length of a tree is the sum of all lengths of all paths from each external node to the root

• Binary Tree: Property 3

  

  The external path length of a binary tree with N internal nodes (and by Property 1, N+1 external nodes) is exactly 2N greater than its internal path length Proof: can be done with induction. I won't prove it because this is a little known fact and rarely used.

• Binary Tree: Property 4 - Min and Max Height of Binary tree

  

  The height of a binary tree with N internal nodes (and by Property 1, N+1 external nodes) is at least lg(N) and at most N+1 Proof: The worst case ("tallest") binary tree with N internal nodes is a tree where the internal nodes for a straight line. The height of this "tallest" possible tree is N+1 The best case ("shortest") binary tree with N internal nodes is every level of the tree is full. There is max 1 (20) internal node at level 0 There are max 2 (21) internal node at level 1 There are max 4 (22) internal node at level 2 ... There are max 2k nodes at level k. So a tree that has a height h can have 1 + 2 + 4 + .. + 2h-1 = 2h -1         internal nodes We are given that there are N internal nodes. The "shortest" tree that will have N internal nodes will have a height of: 2h - 1 = N h = lg(N+1)