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First, consider the tree on the left.  On the right, the nine nodes of the tree have been assigned numbers  from the set $\left\{1, 2,\ldots,9\right\}$ so that for every node, the numbers in its left subtree and right subtree lie in disjoint intervals (that is, all numbers in one subtree are less than all numbers in the other subtree). How many such assignments are possible? Hint: Fix a value for the root and ask what values can then appear in its left and right subtrees.

1. $2^{9}=512$
2. $2^{4}.3^{2}.5.9=6480$
3. $2^{3}.3.5.9=1080$
4. $2^{4}=16$
5. $2^{3}.3^{3}=216$
in DS
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how to solve this question ?

Option is B.

for every node -all numbers in one subtree are less than all numbers in the other subtree .

Firstly chose a value for root $- 9$ elements = $\mathbf{9}$ ways

Now, we hv $\mathbf{8}$ elements left - we hv to chose $\mathbf{3}$ for left subtree & $\mathbf{5}$ for right subtree.

Note: Here we can either chose $3$ nodes from beginning or end out of 8 elements we have ! $= \mathbf{2}$ ways

Now,we hv $3$ elements for left subtree & $5$ for right(Consider subtrees of subtree).

Left Subtree :

whatever way we place , always one side is smaller than other {$6$ is smaller than $8$ in above example given in question} so, total ways $= \mathbf{3!}$ {three places put one by one} $=\mathbf{6}$ ways

Right Subtree :

Right subtree has two more sub-trees ,so that elements on one side should be smaller than other **

Steps :

1. Select one element for root $=\mathbf{5}$ ways
2. $4$ elements left ,Select one element for left $=\mathbf{2}$ ways {Either we can chose from left or right}
3. $3$ elements left, for right subtree $=\mathbf{3!}$ ways $=\mathbf{6}$ ways

Total ways $= 9*2* 3! * 5 * 2 * 3! = 2^4* 3^2 * 5 * 9 = 6480 =$ B (Ans)

by Boss (15.9k points)
edited
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i have a doubt regarding this as u said there are 9 ways to select root suppose i select 1 as root then how we construct left sub tree bcoz after selecting root as 1 there is no other elements smaller than 1 but we need exactely 3 elemants for left sub tree. plzz explain
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all numbers in one subtree are less than all numbers in the other subtree
+4

root is not a matter here

Here only we have to maintain "all numbers in one subtree are less than all numbers in the other subtree"

See here in question 7 is root.

That is not an issue for this question.

But all node of left subtree i.e.(6,9,8) must be greater than all node of right subtree (1,2,3,4,5)

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nice explanation.....
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srestha

What are 9 ways for root?m not getting  Shashi Kant Verma diagram.

+1
All the 9 numbers can be there for the root.

Suppose 8 is chosen as root so from the left numbers 1 2 3 4 5 6 7 9 we have to fill the left and right trees such that left tree values are less than right or vice versa.

We can choose 123 or 679 for the left structure. If we choose  123 then 45679 have to be arranged on right structure.

Hope this helps.
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gari   if 1 choosen as a root?

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Root can be 1 also.  If it is chosen as 1 then the left choices are 23456789. Now our aim is to assign the numbers in such a way that left subtree Values are less than right subtree values or vice versa . There is no relation with the root being less or more than any of the subtree.

Now from 23456789 either 234 or 789 as left. If 789 as left then 23456 to be assigned to right.
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Fix a value for the root and ask what values can then appear in its left and right subtrees

supose if we choose 1 as root ,we can also think like this as in its left part (23456789) and at right nothing .it also satisfies condition.

+1
we have to answer according to the tree structure given in the question.
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Jst Superb ...
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himanshu sir great explanation
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Great explaination
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Here we cannot change tree structure. right??

Means we cannot do $4$ nodes in left subtree and $4$ node in right subtree.

That is not possible, though it is not mentioned in question.

How to understand tree structure will remain same every time. Based on concepts of Combination

by (249 points)
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Could You please explain it in detail.. ?? i am not getting 3 ways and 5 ways part, what are the numbers ??
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why there are 2 ways and 2 ways in left subtree it should be 2 ways and one way respectively