I think somewhere on beta the players figured out the bronze gives a slight advantage between the higher chance at lanterns and you get an extra tree per chest open.
Hmm ... this is an important point that quite invalidates the previous calculations. The average number of lanterns from trees should be directly subtracted from the cost of each chest, which benefits bronze more than silver or gold -- unless the number of tree lanterns depends on the cost of the chest, which it may. Also, the probability of winning the daily are better than .07, .13, and .25, because those numbers should only be applied to the outcomes that don't include winning lanterns. These are
bronze: .07 / (1 - .20) = .0875
silver: .13 / (1 - .15) = .1529
gold: .25 / (1 - .12) = .2841
But you also have to subtract from the cost of the chest the average number of extra lanterns that you win. So assuming that you get on average 3 lanterns per tree, the costs of the chests are
bronze: 40 - 3 - (70 - 40) * .20 = 31
silver: 70 - 3 - (120 - 70) * .15 = 59.5
gold: 120 - 3 - (200 - 120) * .12 = 107.4
Hmm ... that's not right because each time you win lanterns, you get another chance to win lanterns, so it's actually .20**1 + .20**2 + .20**3 ... which I believe is .20/(1 - .20), so the correct costs are
bronze: 40 - 3 - (70 - 40) * .20/(1 - .20) = 29.5
silver: 70 - 3 - (120 - 70) * .15/(1 - .15) = 58.1
gold: 120 - 3 - (200 - 120) * .12/(1 - .12) = 106.1
So starting with 600 lanterns the probability of winning the daily are approximately (not quite accurate because you can't buy a fraction of a chest):
bronze: 1- (1 - .07 / (1 - .20))**(600/(40 - 3 - (70 - 40) * .20/(1 - .20)) = .845
silver: 1- (1 - .13 / (1 - .15))**(600/(70 - 3 - (120 - 70) * .15/(1 - .15))) = .819
gold: 1- (1 - .25 / (1 - .12))**(600/(120 - 3 - (200 - 120) * .12/(1 - .12))) = .849
If I got this right (what are the odds of that?), gold is still best, but narrowly.