The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum number of nodes in a binary tree of height $$h$$ is:

Consider a weighted undirected graph with positive edge weights and let $$uv$$ be an edge in the graph. It is known that the shortest path from the source vertex $$s$$ to $$u$$ has weight 53 and the shortest path from $$s$$ to $$v$$ has weighted 65. Which one of the following statements is always true?

Let $$G$$ be the non-planar graph with minimum possible number of edges. Then $$G$$ has

Consider a weighted complete graph $$G$$ on the vertex set $$\left\{ {{v_1},\,\,\,{v_2},....,\,\,\,{v_n}} \right\}$$ such that the weight of the edge $$\left( {{v_i},\,\,\,\,{v_j}} \right)$$ is $$2\left| {i - j} \right|$$. The weight of a minimum spanning tree of $$G$$ is