Which of the following sequences could not be the sequences of nodes examined in a binary search tree?

Question

Suppose that we have numbers between $1$ and $1000$ in a binary search tree and want to search for the number $363$. Which of the following sequences could not be the sequences of nodes examined

a) 924,220,911,244,898,258,362,363$

b) 925,202,911,240,912,245,363

c) 2,399,387,219,266,382,381,278,363

d) 935,278,347,621,299,392,358,363

e) 925, 202, 911, 240, 912, 245, 363

I know that $d$ and $e$ are not valid sequences, my professor told us so. I know it has something to do the range between jumping from node to node, which has something to do with the following property of BST's

Let $x$ be a node in a binary search tree. If $y$ is a node in the left subtree of $x$, then $y.key \leq x.key$. If $y$ is a node in the right subtree of $x$, then $y.key \geq x.key$.

I just don't see it, been trying to simulate the path sequence with no clear breakthrough.

Answer 1

Actually the test is quite easy. If you consider the subsequence of numbers greater than the final number $363$, this sequence is supposed to decrease.

Thus, in the case of $e: 925, 202, 911, 240, 912, 245, 363$, the subsequence is $925, 911, 912, 363$ and we see the problem.

Why is that so? If we see a number $x$ larger than the target $363$, we are supposed to move to the left subtree of $x$ where all numbers will be smaller than $x$. Thus also all numbers we will see from there will be smaller. As a consequence the numbers larger than $363$ will decrease.

Similarly, the numbers smaller than $363$ will increase. That is why $d$ fails: $278,347,299,358,363$.