Question

A binary tree is $f u l l$ if all of its vertices have either zero or two children. Let $B_{n}$ denote the number of full binary trees with $n$ vertices. (a) By drawing out all full binary trees with $3,5,$ or 7 vertices, determine the exact values of $B_{3}, B_{5},$ and $B_{7},$ Why have we left out even numbers of vertices, like $B_{4} ?$ (b) For general $n,$ derive a recurrence relation for $B_{n^{*}}$ (c) Show by induction that $B_{n}$ is $\Omega\left(2^{n}\right)$.

Name: Problem 13, Chapter 1: Algorithms
Uploaded: 2020-08-13T04:31:05-08:00
Duration: 6 min 1 s
Channel: Chris Trentman

   A binary tree is $f u l l$ if all of its vertices have either zero or two children. Let $B_{n}$ denote the number of full binary trees with $n$ vertices.
(a) By drawing out all full binary trees with $3,5,$ or 7 vertices, determine the exact values of $B_{3}, B_{5},$ and $B_{7},$ Why have we left out even numbers of vertices, like $B_{4} ?$
(b) For general $n,$ derive a recurrence relation for $B_{n^{*}}$
(c) Show by induction that $B_{n}$ is $\Omega\left(2^{n}\right)$.

Algorithms

Sanjoy Dasgupta,… 1st Edition

Chapter 1, Problem 13

Instant Answer

Solved by Expert Chris Trentman

08/13/2020

Step 1

For $B_3,$ there is only one possible full binary tree: ``` o / \ o o ``` So $B_3 = 1.$ For $B_5,$ there are two possible full binary trees: ``` o o / \ / \ o o o o / o ``` So $B_5 = 2.$ For $B_7,$ there are five Show more…

Show all steps

Thanks for your feedback!

A binary tree is $f u l l$ if all of its vertices have either zero or two children. Let $B_{n}$ denote the number of full binary trees with $n$ vertices. (a) By drawing out all full binary trees with $3,5,$ or 7 vertices, determine the exact values of $B_{3}, B_{5},$ and $B_{7},$ Why have we left out even numbers of vertices, like $B_{4} ?$ (b) For general $n,$ derive a recurrence relation for $B_{n^{*}}$ (c) Show by induction that $B_{n}$ is $\Omega\left(2^{n}\right)$.