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I haven't tried this, but provide an answer for each please.

1) When Buzzwole knocks out a Pokemon, will it's attack rise, or defense; or is a random selection?
2) Since Guzzlord's best stat is HP (followed by attack), when Beast Boost activates, will it's attack rise? If not, what happens?

~ Just curious

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OP asks for clarification of specific examples, so I allow it.

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Well, it's important to note that Beast Boost functions on the values of the Pokemon's actual stats, rather than their base stats. However, I'll answer your question on the assumption that a theoretical Buzzwole has equal Attack and Defense and that a theoretical Guzzlord's highest stats are HP and Attack.

In your first case:

For determining the highest stat, Beast Boost does not take into account stat stages, held items, or reductions due to status conditions; however, it does consider the effects of Power Split, Guard Split, Power Trick, Wonder Room, and Speed Swap. In the case of a tie between multiple stats, it increases one stat, prioritized in this order: Attack, Defense, Special Attack, Special Defense, and Speed.

Thus, theoretical Buzzwole's Attack stat would be risen by Beast Boost.

In your second case,

When a Pokémon with Beast Boost causes another Pokémon (including allies) to faint through direct damage, its highest stat (other than HP) rises by one stage. If its highest stat is already at +6 stages, Beast Boost will not activate (not even in the case of a tie of two or more stats).

Therefore, theoretical Guzzlord's Attack stat would be risen by Beast Boost, if it is theoretical Guzzlord's second highest stat.

Hope I helped. :)
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These UBs increase (These) stats:

Nihilego (Special Defense)
Buzzwole (based on the Source, it will increase Attack)
Pheromosa (Speed)
Xurkitree (Special Attack)
Celesteela (Special Attack)
Kartana (Attack)
Guzzlord (Attack)
That's not necessarily true. For example, players often intentionally distribute their Celesteelas' EVs so beast boost boosts special defense.
Thanks Indigo, it's clear to me now
I based it off of base stats, not EVs. When comparing stats, I DO NOT base it off of EVs.
Beast boost does base off EVs, so not basing off EVs is wrong.