1. (10 points) Fundamentals of Trees: (a) Show that if a...

1. (10 points) Fundamentals of Trees: (a) Show that if a connected graph has fewer edges than vertices, then it must be a tree. (b) What is the maximum number of vertices of an $m$-ary tree of height $h$? (c) Let $T$ be any fixed tree. We say that a vertex $v$ of $T$ is a center of $T$ if making $v$ the root of $T$ causes $T$ to have the smallest possible height. Show that $T$ can have at most two centers. (d) Prove that the average height of a leaf in a binary tree is at least the base-two logarithm of the number of leaves.

Transcript

00:01 So we have a weighted graph here, and we want to ask some questions relating to minimum spanning trees.

00:06 In particular, part 3 has us using krusekal's algorithm.

00:09 So i'm going to, for part 1, finding the cost of its minimum spanning tree, find the minimum spanning tree with krusekal's algorithm.

00:18 Step 1 is going to be to look at the lightest node that does not close a loop in our existing tree.

00:27 It's going to be either this one or that one.

00:30 In particular, adding them both will not cause a loop, so we're not going to worry about that.

00:39 Then for number 2, for our next node, we're done with...

00:44 I can't speak.

00:48 The least weight that we see on here now is 2, so we can add this path here, but then this 2 is blocked off because it would close a loop.

01:02 We're now out of 2s, so we have to move on to the 3s.

01:06 And then both of these 3s are fair game, so we're good with that.

01:11 Then 4, we can do a 4 here, and then we would have 5.

01:19 What we'll note here is that this 5 would close a loop.

01:23 At least, i should color it in red.

01:25 That's not allowed, it would make a loop here, this triangle.

01:28 And then this 5 would also make a loop, the e, b, c, g, f, e.

01:32 So those are all off limits, and we're left only with this 5, which brings us to d and also creates our spanning tree.

01:43 You can ignore those ifs there, and we want to find the weight.

01:47 Going through, that's going to be 1 plus 2 plus 4 plus 5 plus 1 plus 3 plus 3 is our total weight.

01:57 1 plus 2 is 3, plus 4 is 7, plus 5 is 12, 13, 16, 19.

02:05 So our total spanning weight is 19...

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