<p>Given an integer array <code>nums</code> and two integers <code>lower</code> and <code>upper</code>, return <em>the number of range sums that lie in</em> <code>[lower, upper]</code> <em>inclusive</em>.</p>

<p>Range sum <code>S(i, j)</code> is defined as the sum of the elements in <code>nums</code> between indices <code>i</code> and <code>j</code> inclusive, where <code>i &lt;= j</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [-2,5,-1], lower = -2, upper = 2
<strong>Output:</strong> 3
<strong>Explanation:</strong> The three ranges are: [0,0], [2,2], and [0,2] and their respective sums are: -2, -1, 2.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [0], lower = 0, upper = 0
<strong>Output:</strong> 1
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-2<sup>31</sup> &lt;= nums[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>-10<sup>5</sup> &lt;= lower &lt;= upper &lt;= 10<sup>5</sup></code></li>
	<li>The answer is <strong>guaranteed</strong> to fit in a <strong>32-bit</strong> integer.</li>
</ul>

class SegmentTreeNode:
	def __init__(self,low,high):
		self.low = low
		self.high = high
		self.left = None
		self.right = None
		self.cnt = 0        
        
class Solution: 
    def _bulid(self, left, right): 
        root = SegmentTreeNode(self.cumsum[left],self.cumsum[right])
        if left == right:
            return root
        
        mid = (left+right)//2
        root.left = self._bulid(left, mid)
        root.right = self._bulid(mid+1, right)
        return root
    
    def _update(self, root, val):
        if not root:
            return 
        if root.low<=val<=root.high:
            root.cnt += 1
            self._update(root.left, val)
            self._update(root.right, val)
                  
    def _query(self, root, lower, upper):
        if lower <= root.low and root.high <= upper:
            return root.cnt
        if upper < root.low or root.high < lower:
            return 0
        return self._query(root.left, lower, upper) + self._query(root.right, lower, upper)
        
    # prefix-sum + SegmentTree | O(nlogn)      
    def countRangeSum(self, nums: List[int], lower: int, upper: int) -> int:
        cumsum = [0]
        for n in nums:
            cumsum.append(cumsum[-1]+n)
            
        self.cumsum = sorted(list(set(cumsum))) # need sorted
        root = self._bulid(0,len(self.cumsum)-1)
        
        res = 0
        for csum in cumsum:
            res += self._query(root, csum-upper, csum-lower)
            self._update(root, csum)        
        return res
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