You are given a weighted tree T.(As a reminder, a tree T is a graph that is connected and contains no cycle.) Each node of the tree T has a weight, denoted by w(v). You want to select a subset of tree nodes, such that weight of the selected nodes is maximized, and if a node is selected, then none of its neighbors are selected.

A mathematician applies Prim’s Algorithm to find a minimum spanning tree for the weighted graph starting at vertex D. The order of the edges picked so far is AD, AB, BE, and EF. The next edge selected when applying Prim’s Algorithm should be BF FG CG CE   What is the total weight of the tree that uses the edges DE, CE, CG, FG and BF? 93 92 91 90   The tree described in problem #8 is a spanning True False

Based on Theorem 4.4.2 P.165, the average height of a leaf in a binary tree with 128 leaves is at least

Question 1: Construct the labeled tree T with Prüfer code 1564263.

Given 8 red nodes, design a red-black tree with (a) minimum height 4 (b) maximum height o.

Let f h denotes the minimum number of nodes in an AVL tree of height h, then prove that 1+v5 fh2c•6" where ø = and c is a constant.

. In a B+-tree, there can be at most 4 search-key values in each node. Assume that the height of the B+-tree is 2. The height of the tree is the length of the path from the root to a leaf node. Let N1 be a node in the tree. N1 is neither a leaf nor the root. What is the minimum number of children that N1 can have?

Question 2 Prove that any tree on n nodes has n- 1 edges.

Show that if a tree with n vertices has two vertices of degree 3 then it must have at least 4 vertices of degree 1