For the branch, I'd probably go with a Girth Hitch (i.e., Cow on a Loop). While I'm tying a Girth, I might even go ahead with a Prusik instead.
For the tire, the knot there will be used as a handle when a person is sitting or standing on the tire, unless you make the knot way up there above the tire. I'd probably use a Double Bowline, as you originally suggested. The working end dangles downward out of the way, and the Double Bowline is trustworthy in my experience. I'd probably combo that Loop with a Cow Hitch on the tire (resulting in a Girth Hitch again).
Other viable options include a Figure 8 Follow Through or a Yosemite Bowline. I imagine these knots would be more tamper resistant than a regular Bowline and neater than a Zeppelin Loop. A Zeppelin Loop would leave a strand of rope hanging sideways, unless you awkwardly fixed the working end to the standing end above.