How to find the width of the river?
You can have the width of the river without crossing it or the height of the tree without climbing, using school geometry or (trigonometry).
We assume the river is of constant width at the place and its banks are parallel at the place we do this experiment.
Identify a tree or any object on the other bank of the river straight across the river.We call this object A. Let your position on this bank be O. Now move along this bank seeing object on the other side of the bank till you reach a point B from where BA makes an angle of 45 degree with BO.
Now measure BO or OB on this bank (at least by steps if have nothing to measure). Then BO is the width of the river. Why? Actually AOB is a right angled isosceles triangle, with angle AOB= 90 deg, OBA = 45 degree and therefore BAO =45. So, BO = AO, which is the width of the river.