The trees in a peach orchard were arranged in a square array with $N$ columns and $N$ rows. To enlarge the orchard by one row and one column to make a larger square, the owner bought and planted 29 more peach trees. How many trees are in the orchard after the enlargement?

Let's denote the number of columns and rows in the original peach orchard as \( N \). Therefore, the total number of trees in the original orchard can be expressed as:

\[
N^2
\]

When the owner enlarges the orchard by adding one additional row and one additional column, the dimensions become \( N+1 \) for both the width and the height. Consequently, the new total number of trees becomes:

\[
(N+1)^2
\]

The problem states that 29 additional trees were bought and planted during this enlargement. This gives us the relationship:

\[
(N+1)^2 = N^2 + 29
\]

We can expand the left-hand side of the equation:

\[
N^2 + 2N + 1 = N^2 + 29
\]

Now, we can simplify this equation by subtracting \( N^2 \) from both sides:

\[
2N + 1 = 29
\]

Next, we isolate \( N \) by subtracting 1 from both sides:

\[
2N = 28
\]

Dividing both sides by 2, we find:

\[
N = 14
\]

Now that we know \( N \), we can determine the original number of trees in the orchard:

\[
N^2 = 14^2 = 196
\]

After the enlargement, the new total number of trees is calculated as follows:

\[
(N+1)^2 = 15^2 = 225
\]

Thus, the total number of trees in the orchard after the enlargement is:

\[
\boxed{225}
\]