number-of-ways-to-reorder-array-to-get-same-bst

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Algorithms

Given an array <code>nums</code> that represents a permutation of integers from <code>1</code> to <code>n</code>. We are going to construct a binary search tree (BST) by inserting the elements of <code>nums</code> in order into an initially empty BST. Find the number of different ways to reorder <code>nums</code> so that the constructed BST is identical to that formed from the original array <code>nums</code>.

<ul>
	<li>For example, given <code>nums = [2,1,3]</code>, we will have 2 as the root, 1 as a left child, and 3 as a right child. The array <code>[2,3,1]</code> also yields the same BST but <code>[3,2,1]</code> yields a different BST.</li>
</ul>

Return the number of ways to reorder <code>nums</code> such that the BST formed is identical to the original BST formed from <code>nums</code>.

Since the answer may be very large, return it modulo <code>109 + 7</code>.

&nbsp;
Example 1:
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/12/bb.png" style="width: 121px; height: 101px;" />
<pre>
Input: nums = [2,1,3]
Output: 1
Explanation: We can reorder nums to be [2,3,1] which will yield the same BST. There are no other ways to reorder nums which will yield the same BST.
</pre>

Example 2:
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/12/ex1.png" style="width: 241px; height: 161px;" />
<pre>
Input: nums = [3,4,5,1,2]
Output: 5
Explanation: The following 5 arrays will yield the same BST: 
[3,1,2,4,5]
[3,1,4,2,5]
[3,1,4,5,2]
[3,4,1,2,5]
[3,4,1,5,2]
</pre>

Example 3:
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/12/ex4.png" style="width: 121px; height: 161px;" />
<pre>
Input: nums = [1,2,3]
Output: 0
Explanation: There are no other orderings of nums that will yield the same BST.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>1 &lt;= nums.length &lt;= 1000</code></li>
	<li><code>1 &lt;= nums[i] &lt;= nums.length</code></li>
	<li>All integers in <code>nums</code> are distinct.</li>
</ul>