<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 segTree:
    def __init__(self, low, high):
        self.low = low
        self.high = high
        self.left = None
        self.right = None
        self.cnt = 0

class Solution:
    def build(self, l, r):
        root = segTree(self.prefix[l], self.prefix[r])
        if l == r:
            return root

        mid = (l + r) // 2
        root.left = self.build(l, mid)
        root.right = self.build(mid + 1, r)
        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)
        return
    
    def query(self, root, lower, upper):
        if lower <= root.low and root.high <= upper:
            return root.cnt
        
        if root.high < lower or upper < root.low:
            return 0

        return self.query(root.left, lower, upper) + self.query(root.right, lower, upper)

    def countRangeSum(self, nums: List[int], lower: int, upper: int) -> int:
        prefix = [0]
        for num in nums:
            prefix.append(prefix[-1] + num)
        
        self.prefix = sorted(list(set(prefix)))
        root = self.build(0, len(self.prefix) - 1)

        ans = 0
        for pre in prefix:
            ans += self.query(root, pre - upper, pre - lower)
            self.update(root, pre)
            
        return ans


        
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