In this problem we are going to deal with a type of weighted binary tree in which each node contains one positive integer and has either no sub$-$trees or exactly one left sub$-$tree and one right sub$-$tree. On the right is a sample weighted binary tree so that you see how to generate those in $E^{A}T{E}_{E}x.$ By definition, the weight of a binary tree is the sum of all of the integers contained in the nodes of the tree.

In this problem, we consider a special type of weigted binary tree called a balanced weighted binary tree or $BWBT$ defined inductively as follows:

Basis: A single node containing a positive integer and with no sub$-$trees is a BWBT$.$ This node is the root of the BWBT$.$

Inductive rule: If $L$ and $R$ are both BWBTs of equal weight $w,$ then a new BWBT can be constructed by:

adding a new node $n$ whose weight is a positive integer less than or equal to $w,$

adding an edge between $n$ and the root of $L,$ thereby making $L$ the left sub$-$tree of the new tree, and

adding an edge between $n$ and the root of $R,$ thereby making $R$ the right sub$-$tree of the new tree. $n$ is the root of the new tree.

a$.(8$ points$)$ This question asks you to count the number of distinct trees of a given weight win$\{1,2,3,$cdots,$15\}$ and to write down your answer by completing the table below.

Write down your answer in the table below

$\backslash$table$[[$weight of the BWBT$,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[$number of distinct BWBTs with this weight value,,,,,,,,,,,,,,,$]]$

b$.(7$ points$)$ Draw all of the distinct BWBTs whose weight is equal to $14.$

Draw your trees in this box.

$[$This problem is continued on the next page$]$

${?}^1$ The two rightmost trees are considered different because their roots do not have the same left$-$subtrees. Also, they do not have the same right sub$-$trees.

$3$

In this problem we are going to deal with a type of weighted binary tree in which each node contains one positive integer and has either no sub$-$trees or exactly one left sub$-$tree and one right sub$-$tree. On the right is a sample weighted binary tree so that you see how to generate those in $E^{A}FE{x}_x.$ By definition, the weight of a binary tree is the sum of all of the integers contained in the nodes of the tree.

In this problem, we consider a special type of weigted binary tree called a balanced weighted binary tree or $BWBT$ defined inductively as follows:

Basis: A single node containing a positive integer and with no sub$-$trees is a BWBT$.$ This node is the root of the BWBT$.$

Inductive rule: If $L$ and $R$ are both BWBTs of equal weight $w,$ then a new BWBT can be constructed by:

adding a new node $n$ whose weight is a positive integer less than or equal to $w,$

adding an edge between $n$ and the root of $L,$ thereby making $L$ the left sub$-$tree of the new tree, and

adding an edge between $n$ and the root of $R,$ thereby making $R$ the right sub$-$tree of the new tree. $n$ is the root of the new tree.

a$.(8$ points$)$ 'I'his question asks you to count the number of distinct trees of a given weight win$\{1,2,3,$cdots,$15\}$ and to write down your answer by completing the table below.

Write down your answer in the table below

$\backslash$table$[[$weight of the BWBT$,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[$number of distinct BWBTs with this weight value,,,,,,,,,,,,,,,$]]$

b$.(7$ points$)$ Draw all of the distinct BWBTs whose weight is equal to $14.$

Draw vour trees in this box.

$[$This problem is continued on the next page$]$

${?}^1$ The two rightmost trees are considered different because their roots do not have the same left$-$subtrees. Also, they do not have the same right sub$-$trees.

$3$