Consider the expression tree shown. Each leaf represents a numerical value, which can either be 0 or 1. Over all possible choices of the values at the leaves, the maximum possible value of the expression represented by the tree is _____.

Let the leaves from left to right be denoted by $x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8$, where each $x_i \in \{0, 1\}$.

We evaluate the subtrees from the bottom up:

1. Left subtree of the root:

   • Left child of $(-)$: $x_1 + x_2$

   • Right child of $(-)$: $x_3 - x_4$

   • Left subtree expression: $(x_1 + x_2) - (x_3 - x_4) = x_1 + x_2 - x_3 + x_4$

2. Right subtree of the root:

   • Left child of $(+)$: $x_5 - x_6$

   • Right child of $(+)$: $x_7 + x_8$

   • Right subtree expression: $(x_5 - x_6) + (x_7 + x_8) = x_5 - x_6 + x_7 + x_8$

3. Root operation $(+)$:

   $$\text{Total Expression} = (x_1 + x_2 - x_3 + x_4) + (x_5 - x_6 + x_7 + x_8)$$

   $$\text{Total Expression} = x_1 + x_2 - x_3 + x_4 + x_5 - x_6 + x_7 + x_8$$

To maximize this expression, we set all positively signed terms to $1$ and all negatively signed terms to $0$:

• $x_1 = 1, x_2 = 1, x_4 = 1, x_5 = 1, x_7 = 1, x_8 = 1$

• $x_3 = 0, x_6 = 0$

Substituting these values: