...
Digital-Logic-Design
October 15, 2023
Mathematical-Reasoning
October 15, 2023
Digital-Logic-Design
October 15, 2023
Mathematical-Reasoning
October 15, 2023

Binary-Trees

Question 46

Let T be a full binary tree with 8 leaves. (A full binary tree has every level full). Suppose two leaves a and b of T are chosen uniformly and independently at random. The expected value of the distance between a and b in T (i.e., the number of edges in the unique path between a and b) is (rounded off to 2 decimal places) _____.

A
5.54
B
1.34
C
4.25
D
3.82
Question 46 Explanation: 
There can be 8 paths between any 2 uniformly & independently chosen leaf nodes.
A node can be chosen twice and the path from that node to itself will be zero.
∴ Path 1 = 0
Similarly,
Path 2 = 2
Path 3 = 4
Path 4 = 4
Path 5 = 6
Path 6 = 6
Path 7 = 6
Path 8 = 6
∴ Expected value = Σ Path length × Probability of selecting path
= 2×1/8 + 4×2/8 + 6×4/8 + 0×1/8
= 1/4 + 1/1 + 3/1 + 0
= 4 + 1/4
= 17/4
= 4.25
Correct Answer: C
Question 46 Explanation: 
There can be 8 paths between any 2 uniformly & independently chosen leaf nodes.
A node can be chosen twice and the path from that node to itself will be zero.
∴ Path 1 = 0
Similarly,
Path 2 = 2
Path 3 = 4
Path 4 = 4
Path 5 = 6
Path 6 = 6
Path 7 = 6
Path 8 = 6
∴ Expected value = Σ Path length × Probability of selecting path
= 2×1/8 + 4×2/8 + 6×4/8 + 0×1/8
= 1/4 + 1/1 + 3/1 + 0
= 4 + 1/4
= 17/4
= 4.25
0 0 votes
Article Rating
Subscribe
Notify of
0 Comments
Inline Feedbacks
View all comments
0
Would love your thoughts, please comment.x
()
x
error: Alert: Content selection is disabled!!