Time Complexity proof for Segment Tree implementation of the ranged sum problem
The claim is that there are at most $2$ nodes which are expanded at each level. We will prove this by contradiction. 

Consider the segment tree given below.



Let's say that there are $3$ nodes that are expanded in this tree. This means that the range is from the left most colored node to the right most colored node. But notice that if the range extends to the right most node, then the full range of the middle node is covered. Thus, this node will immediately return the value and won't be expanded. Thus, we prove that at each level, we expand at most $ 2 $ nodes and since there are $ \log{n} $ levels, the nodes that are expanded are $ \boxed{2 \cdot \log{n} = \Theta\left( {\log{n}} \right)} $