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The following text is snippet from algorithms book.

We could draw the binary trees using rectangular boxes that are customary for linked lists, but trees are generally drawn as circles connected by lines because they are actually graphs. We also do not explicitly draw NULL links when referring to trees because every binary tree with N nodes would require N+1 NULL links.

My question is what does author mean every binary tree with N nodes would require N+1 null links? how author came with N+1 number?

Michael J. Barber
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venkysmarty
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    Try it out yourself. Draw a tree with 1 node (2 nulls), 2 node (3 nulls), 3 nodes (4 nulls), etc. Have you already done this? If so and you are still stuck, we can show you the inductive rule. It is not hard. – Ray Toal Aug 29 '11 at 13:23
  • Another way to look at it: every binary tree with N nodes has N-1 *non*-null links (again, draw some trees and see for yourself). Since there are 2 links per node, you get (2N - (N-1)) = N+1 null links. – comingstorm Aug 29 '11 at 19:38

3 Answers3

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If you have a tree of 1 node, there are 2 null links (left and right on the root). If you add a node to either left or right, you have filled 1 null and added 2 more. This continues on ad infinitum, therefore for each node added you net 1 extra null leaf.

Nick Larsen
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You could prove this by mathematical induction.

Base case

1 node has 2 NULL links - satisfies the property.

Inductive step

Now assume that all trees with n-1 nodes have n NULL links. Then we wish to show, that all trees with n nodes have n+1 NULL links.

Take any tree with n nodes, and pick one of the leaves. Remove this leaf. We now have a tree with n NULL links, per our assumption. If we add the leaf again, we lose one NULL link, but gain two. Thus, we have n - 1 + 2 = n+1 NULL links on the tree with n nodes.

Sebastian Paaske Tørholm
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Observation: Each edge acts as a link. N-1 edges for N nodes.

Totally links: 2N.

Null links: 2N - (N-1) = N+1.

RobinSun
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