A complete binary tree is a graph defined through the following recursive definition. Basis step: A single vertex is a complete binary tree. Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the the graph formed by starting with a root r, and adding an edge from r to each of the vertices r1, r2 is also a complete binary tree. The set of leaves of a complete binary tree can also be defined recursively. Basis step: The root r is a leaf of the complete binary tree with exactly one vertex r. Inductive step: The set of leaves of the tree T built from trees T1, T2 is the union of the sets of leaves of T1 and the set of leaves of T2. The height h(T) of a binary tree is defined in the class. Use structural induction to show that `(T), the number of leaves of a complete binary tree T, satisfies the following inequality `(T) ≤ 2 h(T) .
A complete binary tree is a graph defined through the following recursive definition. Basis step: A single vertex is a complete binary tree. Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the the graph formed by starting with a root r, and adding an edge from r to each of the vertices r1, r2 is also a complete binary tree. The set of leaves of a complete binary tree can also be defined recursively. Basis step: The root r is a leaf of the complete binary tree with exactly one vertex r. Inductive step: The set of leaves of the tree T built from trees T1, T2 is the union of the sets of leaves of T1 and the set of leaves of T2.
The height h(T) of a binary tree is defined in the class. Use structural induction to show that `(T), the number of leaves of a complete binary tree T, satisfies the following inequality `(T) ≤ 2 h(T) .
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