Living Root of the Wildheart - A good druid trinket?

Yes! Absolutely. If you compare this trinket to another role-specific trinket, it might give a bit lower bonus. But, most importantly, it is a versatile 5-in-1 trinket. Every spells and attack would have 3% chance to proc it. If the proc rate or the duration would be increased, then it will overpower trinkets from the similar level.

The blessings are:
In caster form - Cenarion Blessing: +175 damage and healing
In bear form - Ursine Blessing: +4070 armor
In cat form - Feline Blessing: +64 strength
In moonkin form - Lunar Blessing: +209 damage
In tree form - Slyvan Blessing: +326 healing

Caster form:
The use is limited to, resto druids' day-to-day grinding or feral druids' help to heal up now and then.

Bear form:
With a swing timer of 2.5sec and a global cooldown of 1.5sec, 3% proc rate per min, so it should proc 1.92 times/min. Increasing armor by 4070 for 15 sec can be interpreted to 1953.6 armor/sec, which is 355.2 static armor with 5/5 thick hide. This could be an option if Shartuul (or people on your server) refuses to bring you Badge of Tenacity with 308 static armor. From a healer's point of view, this random proc rate won't save mana but increase overhealing. It would be better with some controllable trinkets like Moroe's Lucky Pocket Watch or Scarab of Displacement.

Math:
Proc rate/sec = (60/2.5 + 60/1.5)*3% = 1.92
Increse in armor/sec = 4070*15*1.92/60 = 1953.6
Estimated static armor bonus with Thick hide = 1953.6/550% = 355.2

Cat form:
With a swing timer of 1sec and a global cooldown of 1.5sec, 3% proc rate per min, so it should proc 3 times/min. Increasing strength by 64 for 15sec, can be interpreted into 48 strength i.e. static 96 attack power per sec. This is purely nice. The minor problem is the new Heart of the Wild change when 1 strength would become 2 attack power instead of 2.4 attack power. Therefore, with this trinket, one could focus more on pure attack power and agility.

Math:
Proc rate/sec = (60/1 + 60/1.5)*3% = 3
Increase in strength/sec = 64*15*3/60 = 48 = 96 attack power

Moonkin form:
The best spell rotation to trigger the proc would be IS, MF, Wrath, and then SF when crit or when proc to get the full bonus. Immediately after the proc and approx 2 sec before the blessing fades, MF to get the bonus on the ticks. The static spell damage increase is 62.7 compared to 68.8 from Icon of Silver Crescent. The druid trinket is not as good as that but it is still good.

Math:
Proc rate/sec = 60/1.5*3% = 1.2
Increase in damage = 209*15*1.2/60 = 62.7
Icon of Silver Cresent with a cooldown of 2min:
(43*120+155*20)/120 = 68.8

Tree of Life form:
As it said from description, "spells and attacks" which means melee-ing in tree form would also proc. I tried that this week and it almost doubled my proc rate. Without melee, it gives a static 97.8 healing which is inferior to 133.5 healing from Essence of the Martyr. However, with melee, the static bonus healing increases to 179.3, better than the Essence of the Martyr. If Omen of Clarity is up, the melee attack would grant clearcasting, thus increases the heal per mana also. If you happen to proc when Essence of the Martyr buff is up, then the bonus healing is 707! Cast on the main tank Lifebloom, Regrowth, Rejuvenation, and keep up the Lifebloom, since this around +200 heal/sec can be kept for a long time. ^o^

Math:
Without melee, assuming instant casts only:
Proc rate/sec = 60/1.5*3% = 1.2
Increase in healing = 326*15*1.2/60 = 97.8
With melee (Light's Justice 1.8s) with instant casts:
Proc rate/sec = (60/1.5 + 60/1.8)*3%= 2.2
Increase in healing = 326*15*2.2/60 = 179.3
Essence of the Martyr with a cooldown of 2min:
(84*120+297*20)/120=133.5

The proc rate is a bit low, but if that was increased to 3.5% or the duration extended to 20sec, then it would overpower other trinkets in the same level. In summary, the trinket is very good and versatile, especially for cats and meleeing trees.