Don't treat Storm Debris Cleanup as a generic booking. It is post-weather clearance, where availability rather than price is the real constraint, and that changes what you should ask for. Settle this before you compare prices: putting branches in a hired box wastes the capacity you need for the structural material.
The box has a size and a separate weight cap, and it's the cap, not the size, that decides whether you owe more at pickup. Homeowners after severe weather are the ones most often standing at this decision point, weighing options up to 20 or 30 or 40 yards. What's in the load and how long it stays are the only two variables that move the number.
This job resists sizing in advance because the mix of light and heavy material isn't known until the space is cleared, so expect the estimate to move once the work starts. Weekend and holiday pickup isn't universal -- ask directly if the project timeline runs through one, rather than assuming the schedule matches a weekday drop-off. Pick the number tied to what's actually going in, and let the extra-pull risk be the thing you weigh, not the yardage.