A stroll in the park

Question

Professor Erasmus has returned from his saturday walk in the park. He has counted the number of trees in the park and also the number of lines formed by these trees. Professor Erasmus claims that

• there are exactly 26 trees in the park,
• there are exactly 306 (distinct) lines formed by these trees,
• and that none of these lines does contain four or more trees.

(When the professor speaks of trees on a line, then he means that the centerpoints of the trees lie on a common line.)

We wonder: Has the professor once again made one of his well-known mathematical blunders, or does such a collection of 26 trees indeed exist?

Answer 1

The maximum possible lines is $\sum 25=\frac{25*26}2=325$

The only way to reduce the line count is to put a point into a pre-existing line, which reduces exactly $2$ lines. Instead of $3$ lines $AB$, $BC$ and $CA$, we have a single line $ABC$. We are not allowed to add another point to the line.

Therefore, we can not have an even number of lines, as subtracting $2$s will only result in another odd number. Professor Erasmus has made a blunder.

Answer 2

Expanding on the previous answer - the 325 is right but the second paragraph is wrong - using the 325 lines, you could merge two lines into one - reducing the total amount by 1. Such merger need to be carried out 19 times. By selecting 2 points out of the 26 it seems that you could select 19 points of the remaining 24 to create 19 mergers.