Inspired by hand shaking tree

We start with the root node. It will definitely have $3$ children. In fact, each node(except the root node) will exist as a sibling to $2$ other nodes or as a "triplet". There will be $\frac{100-1}3 = 33$ such triplets. Each of these $33$ triplets will have a unique parent, thus the number of nodes(including root node) with $3$ children will be $33$ . The rest will be leaf nodes and there will be $\boxed{67}$ of them.

After a great struggle, I finally got the possible tree just by following the logic...

I know that I am not so good in drawing and am sorry for that.. :P

p-parents l-leaves Each parent has 3 siblings, some of them are parents except root 3 p=l+p-1 2 p+1=l l+p=3*p+1=100 p=33 l=67

We start with the root as a leaf node. So, we start with 1 node and 1 leaf. In each step, we'll create 3 children to one leaf node. So, in each step, we increase the number of leaf nodes by 2 (3-1 since the new parent becomes an interior node). We need to perform 33 steps and in each step we increase the leaf number by 2 so: 1 + 33x2 = 67