I will explain my approach and I will give you my code.

In this problem we need to keep a structure in each node that can help us to determine the answer for a node from its children. The structure that I used is the following:

struct Node
{
    int sum_left,sum_right,sum_max,sum_total;
    bool indicator;
};

sum_left — the maximum sum of consecutive elements that contains the leftmost element in the range.

sum_right — the maximum sum of consecutive elements that contains the rightmost element in the range.

sum_max — the answer for the interval.

sum_total — the total sum of the elements in the range.

indicator — flag that tells us whether sum_max=sum_total. That is, the whole interval is the one with maximum sum.

Obviously, for leaf nodes indicator=true and sum_right=sum_left=sum_max=sum_total=[The current element in the array].

For non-leaf nodes:

sum_left=(left child).sum_left

if (left child).indicator=true, then sum_left+=(right child).sum_left

sum_right=(right child).sum_right

if (right child).indicator=true, then sum_right+=(left child).sum_right

sum_total=(left child).sum_total + (right child).sum_total

sum_max=max((left node).sum_max,(right node).sum_max,sum_left,sum_right,sum_total,(left node).sum_right+(right node).sum_left)

My code is here.