insert is not working as expected while doing partitioning
Well, with the checks you use for your child tables, it is obvious why such an UPDATE runs into problems.  Adding one to id_ changes it from even to odd or back - the new row does not match the check anymore.

The first trigger is not enough to circumvent this as it does not remove the old row from the child table.  You need a mechanism that does exactly this - delete the old row version and readd the new to either t1 or the matching child.  This trigger has to be defined on t1_odd and t2_even.