Molecular Tree Structures

# Molecular Tree Structures

Trees tend to come up frequently in the science of Chemistry with regards to the structures of certain molecules that have tree like structures. We can model some of these structures with trees by taking each element in a molecule and representing it with a vertex, and a bond between two elements will be represented with an edge. For example, the following molecule CH4 (methane) has the following tree graph:

Notice that this graph is indeed a tree since $\mid V(G) \mid = \mid E(G) \mid - 1$ (5 elements/vertices and 4 bonds/edges).

## Valency of Elements

The valency of an element refers to how many bonds that element can have. For example, the valency of carbon is $1$ since carbon has only $4$ electrons and can accept $4$ more electrons for a full $8$ electron orbital. Note that in these molecular structures, the valency of each element is equal to the degree of that element. For example, since carbon has a valency of $4$, the degree of any carbon atom will always be $4$. The chart below lists some common elements found in trees and their valencies.

Element Atomic Symbol Valency / Degree
Hydrogen H $1$
Carbon C $4$
Nitrogen N $3$
Oxygen O $2$
Chlorine Cl $1$

### Example 1

Show that the chemical molecule C2H6 is a tree structure.

To prove that C2H6 is a tree structure, we must show that $\mid V(G) \mid = \mid E(G) \mid - 1$. We know that this structure will have $8$ vertices/elements ($2$ carbons and $6$ hydrogens), so now we must determine how many edges this structure has. Recall that by The Handshaking Lemma, $\sum_{x \in V(G)} \deg (x) = 2 \mid E(G) \mid$. We know that each carbon has degree/valency $4$, and each hydrogen has degree/valency $1$. Hence it follows that:

(1)
\begin{align} \sum_{x \in V(G)} \deg (x) = 4(2) + 1(6) = 2 \mid E(G) \mid = 14 \\ \mid E(G) \mid = 7. \\ \end{align}

Since $\mid V(G) \mid = 8$ and $\mid E(G) \mid = 7$, then $\mid V(G) \mid = \mid E(G) \mid - 1$, and by equivalence, a graph $G$ representing the molecule C2H6 is a tree.

## Isomers

In chemistry, an isomer of a molecule is another molecule that has the same chemical formula but a different structure. We can evaluate the number of isomers of a molecular formula using graphs.

 Remark: The term "isomer" is not the same as the term "isomorphic" which we defined earlier. Isomers have the same molecular formula, while isomorphisms have the same graphical structure.

### Example 2

Draw all of the isomers for the molecule C4H10.

To determine the number of isomers of C4H10, let's first look at the possible trees we can form on $4$ vertices (carbons), since the placement of the hydrogen atoms afterwards is unimportant right now. We thus obtain the following trees:

Tree 1 Tree 2

So there are exactly $2$ isomers of C4H10. We will omit the structural formula for these isomers as they can easily be obtained by taking the two trees from above and adding hydrogens around the carbon backbone until the valency of each carbon is $4$.