It was a fun unit we did with the kids. They all assumed that the period would vary based on how heavy the "bob" was (in this case the kid on the swing), as well as how big the arc was. They were surprised to find out that neither one made a difference in the period. Here's a pretty good explanation from "YouTube University".
Once the initial angle of the swing gets large (like more than about 45 degrees from vertical), then the angle starts to make a measurable difference. In class the kids could never push a kid on a swing with an angle greater than that. In tree work that could happen, but the difference in the period is still very small. The "big impact" for a climber (pun intended) is not the period, but the velocity on impact. That's where the initial angle combines with the length of the pendulum to make big changes in the velocity from one side of the swing to the other. Between any two given points in a tree, the higher the TIP the lower the max velocity.
Okay, that video was helpful to see the side-by-side comparisons. I was trying to envision the relationship between length and tangential velocity.
To start with, suppose two climbers were in a tree, using the same TIP and standing on the same limb that gives them a displacement from vertical of 30º. They will have the same pendulum length and so, regardless of weight (mass), if they step off at the same time, they will have the same period and swing back to arrive at the limb at the same time. (1:32 in the video) Their velocity should max out at the bottom of the swing (neglecting aerodynamic drag, rope stretch, TIP friction, etc).
If they both move to a lower limb that doubles the length of their pendulum, but keep the angular displacement at 30º, the swing period will increase (but not necessarily double), and again, they will both return to the starting limb (or target limb) at the same time. Their velocity should increase, because they are "falling" farther, but they have a longer arc distance to swing through. So what the video showed me (1:48) is that just because they are traveling faster, that doesn't make up for the farther distance they swing, and so the period of their swing will still be greater.
To me, it always feels like I am going faster in a swing when my TIP is only a few feet above me compared to when it's much longer. Must be the reduced period that I have to "experience" the swing that causes the illusion.
Now, what happens when you change the pendulum length during the swing, like when it shortens as your rope strikes an interfering limb between you and your TIP?