You have two problems.
Fixing the foot bridge that is broken now.
Figuring out source of tension/compression that caused the bowing once it was unbalanced.
Fixing the bridge
Depends how far you want to go with patching this structure for the short term.
The long term fix is to rebuild it with two new beams, and to reuse deck timbers and handrails on the existing ground abutments.
As a short term fix over winter you might attach one/two guy wires on the inside of the curve, and pull them taut to a couple of stout pegs on the bank - Steel warratahs or similar would be adequate as posts. Leave some mechanism so you can add tension to the ropes/straps over time.
Plan view from above
o| |o <-- posts
| \ / |
| \ / |
bank | | bank
| stream |
Another fix would be to reconstruct the bridge using regular X bracing for the entire length, along with sistering/replacing the broken beam.
bank |XXXXXXXX| bank
| stream |
Dealing with the Compression Stress
I'm no engineer, but I've lived in an earthquake-damaged city for ~10 years. Almost every watercourse has suffered accelerated erosion and landform changes. Most bridges have had damage, from their abutments dropping (the bridge is a step up from the roadway) River banks have all slumped.
I suspect from the description that your watercourse has narrowed over the time since the bridge was built. This narrowing is slow, you wouldn't notice it.
Thus, the two banks are pushing on the ends of the bridge, compressing it. When the tree fell and broke one side, the bridge was able to release the pressure sideways.
Road bridges deal with contraction/expansion, but your footbridge is probably secured to each abutment on the bank. The fix is to make one or both ends somewhat "free floating"
Side view from upstream
\ Securing pins /
\ / /
| | |
---|--| |------ < Not secured, free to move
Concrete | / | Concrete
Footing \ / | Footing
00 water 00
The "free-floating" side would be constrained to stop it moving sideways. Or you might prefer to secure both sides using slots to allow for future movement.
Another option is to replace the entire bridge with some pipes to carry water flow, and cover them with soil. This would be a culvert-bridge and could resist the compression from the banks better. Also, concrete/metal pipes are probably a lot cheaper than wood these days, so it might be cheaper. Culvert bridges would potentially carry heavier loads too, but the exact numbers need to be figured out.
Related - plan on doing tree maintenance around your property, at the bare minimum a tree inspection seems wise, which is easiest before leaves come in at the start of Spring. If one tree has broken, others might be simi