Good morning on this soon-to-be-sunny Tuesday.  On Monday afternoon, Mayor Bill de Blasio presented a budget plan for fiscal year 2016.  The Times reporter Nikita Stewart was at City Hall for the unveiling.  “The mayor put public safety at the front and center” of the $77.7 billion plan, she said.  • Fire: $18 million to add ambulance shifts and hire more dispatchers for better ambulance response times.  • Correction: $39.1 million to improve conditions at Rikers Island by increasing the ratio of guards to inmates, among other things.  • Police: $7.3 million for thousands of bulletproof vests, and $10 million more to expand a cadet program.  Still, “some City Council members are saying that he’s not going far enough,” Ms. Stewart said.  Council members, including the speaker, Melissa Mark-Viverito, were disappointed that the budget plan didn’t add 1,000 more police officers, Ms. Stewart said.  Critics also pointed to gaps, namely on spending for libraries and cultural initiatives.  The budget will go through many revisions before it’s formally approved in June.  The mayor deferred a number of question to his “budget folks,” but he seemed “confident,” Ms. Stewart said.  Although he was fighting off a cold, she said, “he was really into his presentation.”  See the city’s summary of the budget plan.  Here’s what else you need to know.  The worst – namely, “freezing drizzle” – seems to have passed.  The clouds are expected to part on a sunny day, with a high of 34.  • Community groups rally to save 17 community gardens on “vacant lots” slated for affordable housing. City Hall steps. 9 a.m.  • A conversation on how to protect the city’s homeless children at the New School. 8:30 to 10:30 a.m. [Livestream]  • Elected officials and anti-hunger campaigners release a report showing “poor participation” in school breakfast programs and call for breakfast at all city schools. City Hall steps. 11 a.m.  • The mayor speaks at the H. F. Guggenheim Symposium on Crime in America at John Jay College. 12:55 p.m.  • The Manhattan district attorney announces charges over a yearslong bribery investigation involving building inspectors and city contractors. Manhattan Family Justice Center. 1 p.m. [Livestream]  • Mayor de Blasio meets the mayor of London, Boris Johnson, at City Hall. 3:30 p.m. …  • And speaks at a preview of the PBS film “The Italian Americans,” at N.Y.U. 6:55 p.m.  • Take a walk through Bronx history without leaving the Mott Haven Library, and see 500 heretofore unseen photos of the borough. 4 p.m. [Free]  • Last day for “Henri Matisse: The Cut-Outs.” If you can’t make it, take a virtual tour.  • The High Line is still closed because of ice.  • Nets at Grizzlies, 8 p.m. (YES). Islanders host Oilers, 7 p.m. (MSG+). Rangers at Maple Leafs, 7:30 p.m. (MSG).  • For more events, see The New York Times Arts & Entertainment guide.  • L.I.R.R., Metro-North, N.J. Transit, Amtrak  • Roads: Check traffic map or radio report on the 1s or the 8s.  • Alternate-side parking: in effect till Thursday (weather permitting).  • Air travel: La Guardia, J.F.K., Newark.  • The stories of the 265 people killed in traffic crashes in New York City in 2014. [WNYC]  • An M.T.A. money collector used his own gun to kill himself at a Greenwich Village subway station on Monday night. [New York Post]  • One person was killed in a pileup involving more than 15 vehicles near Cranbury, N.J., on Monday night. [NJ.com]  • The Court of Appeals, the state’s highest court, is back to full power with two newly appointed judges and a decidedly liberal bent. [New York Times]  • The ex-wife of the man accused of killing Etan Patz testified that he had admitted to having done “something terrible.” [New York Times]  • As of Sunday, no one had been murdered in the city for seven full days, the police reported. [Daily News]  • The city’s new chief medical examiner says a system for tracking bodies as they move from street to morgue will prevent the kind of human errors that have recently plagued the office. [New York Times]  • A well-hidden Google Map feature, nicknamed the drunk train, allows users to see the last train departing to their destination. [Observer]  • Orange fliers for a strip club left on cars in Long Island City made residents think that the police had gone on a ticketing “rampage.” [DNAinfo]  • The besieged Indian builder: Part Three in a series about the people behind the shell companies buying high-end real estate in New York. [New York Times]  • Scoreboard: Heat get to Knicks, 109-95. Bucks shred Nets, 103-97, Oilers slicker than Devils, 2-1.  • For a global look at what’s happening, see Your Tuesday Briefing.  Yesterday, a reader, Alison Hahn, wrote to us with a question.  “Can you tell us why the American flag, specifically in Union Square Park, is flying at half-staff today?”  A website that tracks half-staff American flag notifications around the country did not provide any answers.  So we contacted the people in charge of such things in the city, the Parks Department.  Before long, we heard back from a spokeswoman for the department.  She swiftly put any fears of an untimely death or a missed anniversary to rest.  “Our ops chief said they couldn’t access the flag to raise it,” she said.  “I think due to issues with ice/snow.”  Andy Newman and Joseph Burgess contributed reporting.  New York Today is a weekday roundup that stays live from 6 a.m. till late morning. You can receive it via email.  What would you like to see here to start your day? Post a comment, email us at nytoday@nytimes.com, or reach us via Twitter using #NYToday.  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Tuesday: A budget focused on public safety, some sun, and portraits of the pedestrians killed in the city in 2014.