Statement: Let us define $$a dup subtree in a heap$$ as a heap subtree that does not contain the

path from the root to the node $($the last node in the heap$).$ For example, the above heap tree

$(=26)$ contains $12$ dup subtrees of size $1,[$A$]$ dup subtrees of size $3,[$B$]$ dup subtrees of size

$7,$ and $[$C$]$ dup subtrees of size $15.$ As another example, given a heap tree of size $=1,573,827,$

the number of dup subtrees of size $31$ is $[$D$],$ and the number of dup subtrees of size $127$ is $[$E$].$

Hint: Given a heap tree of size $,$ the number of dup subtrees of size $1$ is $$

$1$

$2$