export exports the variable assignment to child processes of the shell in which the export command was ran. Your command-line environment is the parent of the script's shell, so it does not see the variable assignment.
You can use the . (or source) bash command to execute the script commands in the current shell environment and achieve what you want, e.g.
source ./script.sh
echo "$BASE"
Will produce
/home/develop/trees
The source command, often seen in scripts, is a bash synonym for ., which is part of the POSIX standard (so . is available in dash, for example, but source isn't).
. ./script.sh # identical to "source ./script.sh"
(. script.sh and source script.sh will first look for script.sh in PATH, so it's safer to specify the path to script.sh.)