# Time:  O(n)
# Space: O(h)

import random


class TreeNode(object):
    def __init__(self, val=0, left=None, right=None):
        pass


# bfs, quick select
class Solution(object):
    def kthLargestLevelSum(self, root, k):
        """
        :type root: Optional[TreeNode]
        :type k: int
        :rtype: int
        """
        def nth_element(nums, n, left=0, compare=lambda a, b: a < b):
            def tri_partition(nums, left, right, target, compare):
                mid = left
                while mid <= right:
                    if nums[mid] == target:
                        mid += 1
                    elif compare(nums[mid], target):
                        nums[left], nums[mid] = nums[mid], nums[left]
                        left += 1
                        mid += 1
                    else:
                        nums[mid], nums[right] = nums[right], nums[mid]
                        right -= 1
                return left, right
            
            right = len(nums)-1
            while left <= right:
                pivot_idx = random.randint(left, right)
                pivot_left, pivot_right = tri_partition(nums, left, right, nums[pivot_idx], compare)
                if pivot_left <= n <= pivot_right:
                    return
                elif pivot_left > n:
                    right = pivot_left-1
                else:  # pivot_right < n.
                    left = pivot_right+1
    
        arr = []
        q = [root]
        while q:
            new_q = []
            for u in q:
                if u.left:
                    new_q.append(u.left)
                if u.right:
                    new_q.append(u.right)
            arr.append(sum(x.val for x in q))
            q = new_q
        if k-1 >= len(arr):
            return -1
        nth_element(arr, k-1, compare=lambda a, b: a > b)
        return arr[k-1]