<p>You are given an integer array <code>nums</code> and an integer <code>k</code>.</p>

<p>Find the longest subsequence of <code>nums</code> that meets the following requirements:</p>

<ul>
	<li>The subsequence is <strong>strictly increasing</strong> and</li>
	<li>The difference between adjacent elements in the subsequence is <strong>at most</strong> <code>k</code>.</li>
</ul>

<p>Return<em> the length of the <strong>longest</strong> <strong>subsequence</strong> that meets the requirements.</em></p>

<p>A <strong>subsequence</strong> is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.</p>

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

<pre>
<strong>Input:</strong> nums = [4,2,1,4,3,4,5,8,15], k = 3
<strong>Output:</strong> 5
<strong>Explanation:</strong>
The longest subsequence that meets the requirements is [1,3,4,5,8].
The subsequence has a length of 5, so we return 5.
Note that the subsequence [1,3,4,5,8,15] does not meet the requirements because 15 - 8 = 7 is larger than 3.
</pre>

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

<pre>
<strong>Input:</strong> nums = [7,4,5,1,8,12,4,7], k = 5
<strong>Output:</strong> 4
<strong>Explanation:</strong>
The longest subsequence that meets the requirements is [4,5,8,12].
The subsequence has a length of 4, so we return 4.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,5], k = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong>
The longest subsequence that meets the requirements is [1].
The subsequence has a length of 1, so we return 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>1 &lt;= nums[i], k &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def lengthOfLIS(self, nums: List[int], k: int) -> int:
        max_value = max(nums)
        tree = [0]*(4*max_value)
        def update(node:int,tl:int,tr:int,pos:int,val:int):
            if tl == tr:
                tree[node] = val
                return
            mid = (tl + tr) // 2
            if mid >= pos:
                update(2*node,tl,mid,pos,val)
            else:
                update(2*node+1,mid+1,tr,pos,val)
            tree[node] = max(tree[2*node+1],tree[2*node])
        def query(node:int,tl:int,tr:int,L:int,R:int)->int:
            if tl == L and tr == R:
                return tree[node]
            mid = (tl + tr) // 2
            if mid < L:
                return query(2*node+1,mid+1,tr,L,R)
            elif mid >= R:
                return query(2*node,tl,mid,L,R)
            else:
                return max(query(2*node+1,mid+1,tr,mid+1,R),query(2*node,tl,mid,L,mid))
        maxlength = 0
        for num in nums:
            L = max(num-k,0)
            R = num-1
            val = query(1,0,max_value,L,R) + 1
            maxlength = max(maxlength,val)
            update(1,0,max_value,num,val)
        return maxlength

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